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WHOSE TENDER SYMPATHY HAS HELPED TO
^LIGHTEN THE DARKNESS OP THE*DAY^
DURING WHICH THESE PXoW * v
WERE WRITTEN,

ji the common operations and practices of chymists, almost
letters of the alphabet, without whose knowledge 'tis very
hard for a man to become a philosopher; and yet that knowledge is very far
from being sufficient to make him one."

ROBERT BOYLE. (The Sceptical Chymist.)

PREFACE TO SECOND EDITION.

"The theoretical side of physical chemistry is and will probably remain the
dominant one; it is by this peculiarity that it has exerted such a great in;
fluence upon the neighboring sciences, pure and applied, and on this ground
physical chemistry may be regarded as an excellent school of exact reasoning
for all students of natural sciences." — Arrhenius.

The demand for a second edition of this book has not only
afforded the author an opportunity to thoroughly revise the
original text, but also has made it possible to include such ne\v
material as should properly find a place in an introductory test-
book of theoretical chemistry.

The arduousness of the task of revision and amplification has
been appreciably lightened by the helpful criticisms and valuable
suggestions which have been received from those who have used
the first edition with their classes.

So numerous were the additional topics suggested that the
author found himself confronted with a veritable embarrassment
of riches, and not the least difficult part of his task has been the
attempt to weave in as many of these suggestions as seemed to be
consistent with a well-balanced presentation of the entire subject.

The features which distinguish this edition from the preceding
edition may be briefly summarized as follows: —

1. The necessity of introducing a short chapter on the mod-
ern conception of the atom and its structure involved the
further necessity of including a preliminary chapter treat-
ing of those radioactive phenomena upon which the greater
part of our present atomic theory is based.
The chapter on solids has been practically rewritten:
the space formerly devoted to an outline of crystallog-

vii

viii PREFACE

raphy being devoted in the present edition to a discussion
of the absorption of heat by crystalline solids and the bear-
ing of X-ray spectra on crystalline form.

3. The increasing importance of colloidal phenomena, not
only to the chemist but also to the biologist, to the physician,
and to the technologist, has made it seem desirable to rewrite
the entire chapter devoted to the chemistry of colloids.

4. The Brownian movement and its bearing upon the exist-
ence of molecules has been briefly presented in a separate
chapter in order to emphasize the importance of th^ bril-
liant experimental work of Perrin and others in confirming
the kinetic theory.

5. The chapter treating of electromotive force has been
enlarged so as to include a discussion of some of the more
valuable methods which have been proposed for deter-
mining junction potentials and also to point out several
useful applications of concentration cells.

6. An entirely new chapter in which an attempt has been made
to present the salient facts and more important theories
of photochemistry in succinct form replaces the former
chapter treating of the relations between radiant and
chemical energy.

Among the books to which the author has had frequent re-
course in the preparation of this edition should be mentioned
Rutherford's " Radioactive Substances and their Radiations/' W.
H. and W. L. Bragg's "X-Rays and Crystal Structure/' Freund-
lich's " Kapillarchemie," Perrin's "Les Atomes/' and Sheppard's
" Photochemistry. "

It is with a deep sense of gratitude that acknowledgment is
made to all of those friends who have offered criticisms of the old
edition and suggestions for the new. Special thanks are due
to Dr. W. D. Harkins of the University of Chicago not only for
suggestions but for permission to quote extensively from his
papers, to Dr. J. Howard Matthews of the University of Wis-
consin for numerous helpful suggestions, to Dr. Walter A. Patrick
of Johns Hopkins University for criticisms of the former chapter

PREFACE IX

on colloids and to Mr. John McGavack for his conscientious work
in checking the answers to all of the problems. In the preparation
of the indices of names and subjects the author is indebted to his
wife and to Miss Mary K. Pease who have given valuable assist-
ance in that wearisome and exacting task, To the publishers,
Messrs. John Wiley and Sons, Inc., acknowledgment is made
of their kindness in permitting the use of Pig. 68 taken from
Chamot's "Elementary Chemical Microscopy. "

FREDERICK H. GETMAN.
STAMFORD, CONN.
Aug. 7, 1918.

PREFACE.

"The last thing that we find in making a book is to know what we* must
put first/' — PASCAL.

THE present book is designed to meet the requirements of
classes beginning the study of theoretical or physical chemistry.
A working knowledge of elementary chemistry and physics has
been presupposed in the presentation of the subject, the introduc-
tory chapter being the only portion of the book in which space is
devoted to a review of principles with which the student is assumed
to be already fairly familiar. With the exception of a few para-
graphs in which the application of the calculus is unavoidable,
no use is made of the higher mathematics, so that the book should
be intelligible to the student of very moderate mathematical
attainments. Wherever the calculus has been employed, the
student who is unfamiliar with this useful tool must accept the
correctness of the results without attempting to follow the suc-
cessive operations by which they are obtained.

The contributions to our knowledge in the domain of physical
chemistry have increased with such rapidity within recent years,
that the prospective author of a general text book finds himself
confronted \vith the vexing problem of what to omit rather than
what to include. In selecting material for this book, the author
has been guided in large measure by his own experience in teaching
theoretical chemistry to beginners arid to advanced students.
The attempt has been made to present the more difficult portions
of the subject, such as the osmotic theory of solutions, the laws
of equilibrium and chemical action, and the principles of electro-
chemistry, in a clear and logical manner. While the treatment
of each topic is necessarily brief yet the effort has been made to
avoid the sacrifice of clearness to brevity,

XII PREFACE

The author is fully convinced from his own experience as
as from that of his colleagues, that the complete mastery of the
fundamental principles of the science is best attained through
the solution of numerical examples. For this reason, typical
problems have been appended to various chapters of the book.

Numerous references to original papers have been given through-
out, since the importance of literary research on the part of the
student is conceded by all teachers to be of prime importance.

While a brief account of radioactive phenomena might very
properly be considered to lie within the scope of a general outline
of theoretical chemistry, yet owing to the unparalleled growth
of knowledge in this field during the last decade, the author has
come to believe that a condensed statement of the main facts of
radiochemistry would not be of sufficient value to justify the
effort involved in its preparation.

In the original preparation of his lectures, and in their evolution
into book form, the author has had frequent occasion to consult
Nernst's "Theoretische Chemie," Ostwald's "Lehrbuch der
allgemeinen Chemie," and Van't Hoffs "Vorlesungen ueber
theoretische und physikalische Chemie." Among other books
to which the author is especially indebted are the following: —
Le Blanc's " Lehrbuch der Elektrochemie," Daneel'a "Elek-
trochemie." Text books of Physical Chemistry edited by Sir
William Rarnsay, Bigelow's "Theoretical and Physical Chem-
istry." Jones' "Elements of Physical Chemistry," Reychler-
Kuhn's " Physikalisch-chemische Theorieen," and Whetham's
"Theory of Solution."

In the preparation of the problems the author would record his
indebtedness to Abegg and Sackur's "Physikalisch-chemische
Rechenaufgaben," and to Morgan's "Elements of Physical
Chemistry."

It is a pleasure to acknowledge the valuable assistance rendered
by Dr. Eleanor F. Bliss and Dr. Anna Jonas, who have read and
revised the proof of the paragraphs treating of crystalline form.
The index of titles and names has been prepared by the author's
wife to whose untiring patience its completeness is due. The
author would also record his thanks to those friends whose kindly

PREFACE Xiii

^iticism has helped to remove many blemishes. Finally, the
author would express his appreciation of the kindness of Messrs,
Adam Hilger of London, and Fritz Koehler of Leipzig who have
rendered great assistance by permitting the reproduction of
illustrations of apparatus from their catalogs.

FREDERICK H. GETMAN.

STOCKBRIDGE, MASS.
Aug. 18, 1913.

CONTENTS.

CHAP. PAGE

PREFACE TO SECOND EDITION vii

PREFACE xi

I.^Fundamental Principles 1

^Classification of the Elements 20

IirTThe Electron Theory 31

IV. Radioactivity 42

V.^tomic Structure 57

VI. Gases 72

VII. Liquids : 104

VIII. Solids 153

IX. Solutions 167

X. Dilute Solutions and Osmotic Pressure . , 187

XI. Association, Dissociation and Solvation 225

XII. Colloids 237

XIII. Molecular Reality , 279

XIV. Thermochemistry >. 286

XV. Homogeneous Equilibrium 312

XVI. Heterogeneous Equilibrium X 328

XVII. Chemical Kinetics < 359

XVIII. Electrical Conductance 385

XIX. Electrolytic Equilibrium and Hydrolysis 422

XX. Electromotive Force 446

XXI. Electrolysis and Polarization 492

XXII. Photochemistry 504

INDEX OF NAMES 529

INDEX OF SUBJECTS * 533

THEORETICAL CHEMISTRY.

CHAPTER I.
FUNDAMENTAL PRINCIPLES.

Theoretical Chemistry. That portion of the science of chem-
istry which has for its object the study of the laws controlling
chemical phenomena is called theoretical or physical chemistry.
The first attempt to summarize the more important facts and
ideas underlying the science of chemistry was made by Dalton in
1808 in his "New System of Chemical Philosophy." The birth
of the science of theoretical chemistry may be considered to be
coeval with the appearance of Dalton's epoch-making book.

Theoretical chemistry is concerned with the great generaliza-
tions of chemical science and bears the same relation to chemistry
that philosophy bears to the whole body of scientific truth; it
aims to systematize all of the established facts of chemistry
and to discover the laws governing the various phenomena of
chemical action.

Law, Hypothesis and Theory. The science of chemistry is
based upon experimentally established facts. When a number
of facts have been collected and classified we may proceed to draw
inferences as to the behavior of systems under conditions which
have not been investigated. This process of reasoning by analogy
we term generalization and the conclusion reached we call a law.
It is apparent that a law is not an expression of an infallible truth,
but it is rather a condensed statement of facts which have been
discovered by experiment. It enables us to predict results with-
out recourse to experiment. The fewer the number of cases in
which a law has been found to be invalid, the greater becomes our
confidence in it, until eventually it may come to be regarded as
tantamount to a statement of fact.

2 THEORETICAL CHEMISTRY

Natural laws may be discovered by the correlation of experi-
mentally determined facts, as outlined above, or by means of a
speculation as to the probable cause of the phenomena in question.
Such a speculation in regard to the cause of a phenomenon is
called an hypothesis.

After an hypothesis has been subjected to the test of experiment
and has been shown to apply to a large number of closely related
phenomena it is termed a theory.

In his address to the British Association (Dundee, 1912), Profes-
sor Senier has this to say : " While the method of discovery in chem-
istry may be described generally, as inductive, still ?11 the modes of
inference which have come down to us from Aristotle, analogical,
inductive, and deductive, are freely used. An hypothesis is framed
which is then tested, directly or indirectly, by observation and
experiment. All the skill, and all the resource the inquirer can
command, are brought into his service; his work must be accurate;
and with unqualified devotion to truth he abides by the result,
and the hypothesis is established, and becomes a part of the
theory of science, or is rejected or modified."

Elements and Compounds. All definite chemical substances
are divided into two classes, elements and compounds.

Robert Boyle was the first to make this distinction. He de-
fined an element as a substance which is incapable of resolution
into anything simpler. The substances formed by the chemical
combination of two or more elements he termed chemical com-
pounds. This definition of an element as given by Boyle was
later proposed by Lavoisier and, notwithstanding the vast accumu-
lation of scientific knowledge since their time, the definition re-
mains very satisfactory today.

At the present time we have a group of about eighty substances
which have resisted all efforts to decompose them into simpler
substances. These are the so-called chemical elements. It
should be borne in mind, however, that because we have failed to
resolve these substances into simpler forms of matter, we are not
warranted in maintaining that such resolution may not be effected
in the future.

Recent investigations of the radioactive elements have shown

FUNDAMENTAL PRINCIPLES 3

that they are continuously undergoing a series of transformations,
one of the products of which is the inactive element helium. This
behavior is contrary to the old view that transformation of one
element into another is impossible. At first the attempt was
made to explain it by assuming that the radioactive element was
a compound of helium with another element, but since the radio-
active elements possess all of the properties characteristic of
elements as distinguished from compounds, and find appropriate
places in the periodic table of Mendeteeff, the " compound theory "
must be abandoned. Uranium and thorium, the heaviest ele-
ments known, appear to be undergoing a process of spontaneous
disintegration over which we have no control. The products
of this disintegration have filled the gap in the periodic table
between thorium -and lead with about thirty new elements, each
of which is in turn undergoing transformations similar to those
of the parent elements. Professor Soddy * says: "In spite cf
the existence at one time of a vague belief (a belief which has no
foundation), that all matter may be to a certain extent radioactive,
just as all matter is believed to be to a certain extent magnetic, it
is recognized today that radioactivity is an exceedingly rare prop-
erty of matter."

Notwithstanding these remarkable discoveries, we may still hold
to the idea of an element as suggested by Boyle and Lavoisier.
Professor Walker says: " The elements form a group of substances,
singular not only with respect to the resistance which they offer
to decomposition, but also with respect to certain regularities dis-
played by them and not shared by substances which are designated
as compounds."

Law of the Conservation of Mass. In 1774, as the result of
a series of experiments, Lavoisier established the law of the con-
servation of mass which may be stated as follows: In a chemical
reaction the total mass of the reacting substances is equal to the total
mass of the products of the reaction. It is sometimes stated thus: —
the total mass of the universe is a constant; but this form of state-
ment is open to the objection that we have no means of verification;
it is a statement of a fact which transcends our experience.

* Chemistry of the Radio-Elements, p. 2.

THEORETICAL CHEMISTRY

The law of the conservation of mass has been subjected to most
rigid investigation by Landolt * in a series of experiments extend-
ing over a period of fifteen years.

The reacting substances, AB and CD,
were placed in the two arms of the in-
verted U-tube shown in Fig. 1 which was
then sealed at S and the whole weighed
upon an extremely sensitive balance. The
vessel was then inverted when the following
reaction took place: —

CD-+AD + CB.

Fig. 1.

When the reaction was complete and suf-
ficient time had elapsed to allow the vessel
to return to its original volume (this some-
times required nearly three weeks), it was
weighed again using every precaution to
avoid errors and any gain or loss in weight
noted.

Landolt concluded from the thirty or more reactions which he
studied that the gain or loss in weight was less than one ten-
millionth of the total weight.f

Law of Definite Proportions. The enunciation of the law of
the conservation of mass and the introduction of the balance into
the chemical laboratory marked the beginning of a new era in the
history of chemistry, — the era of quantitative chemistry. As
the result of painstaking experimental work, Richter and Proust
announced the law of definite proportions about the beginning
of the nineteenth century. This law may be expressed thus:
A definite chemical compound always contains the same elements
united in the same proportion by weight.

Shortly after the enunciation of this law its truth was questioned
by the French chemist Berthollet.J From the results of a series

* Zeit. phys. chem., 12, 1 (1893); 55, 589 (1906).

t An excellent summary of this important investigation will be found io
the Journal de Chimie physique, 6, 625 (1908).
J Essai de statique chimique (1803) . *

FUNDAMENTAL PRINCIPLES 5

of brilliant experiments, he became convinced that chemical reac-
tions are largely controlled by the relative amounts of the react-
ing substances. As we shall see later, he really foreshadowed the
work of Guldberg and Waage who were the first to correctly for-
mulate the influence of mass on a chemical reaction. Berthollet
argued that when two elements unite to form a compound, the
proportion of one of the elements in the compound is conditioned
solely by the amount of that element which is available. This
led to the celebrated controversy between Berthollet and Proust
which finally resulted in the establishment of the latter's original
statement. Subsequent investigation has only strengthened our
faith in the law of definite proportions.

Law of Multiple Proportions. Elements are known to unite
in more than one proportion by weight. Dalton analyzed the
two compounds of carbon and hydrogen, methane and ethylene,
and found that the ratio of the weights of carbon to hydrogen in
the former was 6 : 2 while in the latter it was 6:1. That is, for the
same weight of carbon, the weights of the hydrogen in the two
compounds were in the ratio 2:1.

A large number of compounds were examined and similar
simple ratios between the masses of the constituent elements
were found. As a result of these observations, Dalton * formu-,
lated in 1808 the law of multiple proportions, as follows: When
two elements unite in more than one proportion, for a fixed mass of
one element the masses of the other element bear to each other a simple
ratio. Notwithstanding the fact that Dalton was a careless
experimenter the subsequent investigations of Marignac and
others have established the validity of his law.

Law of Combining Proportions. Dalton pointed out that it is
possible to assign to every element a definite relative weight with
which it enters into chemical combination. He observed that
the weights or simple multiples of the weights of the different
elements which unite with a given weight of a definite element,
represent the weights of the different elements which combine
with each other. The weights of the elements which combine
with each other are termed their combining weights. This com-

* A New System of Chemical Philosophy (1808).

6 THEORETICAL CHEMISTRY

prehensive law of chemical combination may be stated as follows:
Elements combine in the ratio of their combining weights or in
simple multiples of this ratio. It will be observed that this law
really includes the law of definite and the law of multiple pro-
portions.

If we assume the combining weight of hydrogen to be unity,
the combining weights of chlorine, oxygen and sulphur will be
35.5, 8 and 16 respectively. These numbers represent the ratios
in which the elements substitute each other in chemical com-
pounds. Hydrochloric acid, for example, contains 35.5 parts by
weight of chlorine to 1 part by weight of hydrogen and when
oxygen is substituted for chlorine, forming water, the new com-
pound contains 8 parts by weight of oxygen to 1 part by weight
of hydrogen. Similarly, if the oxygen be substituted by sulphur,
forming hydrogen sulphide, there will be found 16 parts by weight
of sulphur to 1 part by weight of hydrogen. We may say, then,
that 35.5 parts of chlorine, 8 parts of oxygen and 16 parts of sul-
phur are equivalent.

A chemical equivalent may be defined as the weight of an element
which is necessary to combine with or displace 1 part by weight
of hydrogen.

The Atomic Theory. In very early times two different views
were entertained by opposing schools of Greek philosophers as to
the mechanical constitution of matter. According to the school
of Plato and Aristotle, matter was thought to be continuous
within the space it appears to fill and to be capable of indefinite
subdivision. According to the other school, first taught by
Leucippus, and afterwards by Democritus and Epicurus, matter
was considered to be made up of primordial, extremely minute
particles, distinct and separable from each other but in themselves
incapable of division. These ultimate particles were called atoms
(i^TOfios), signifying something indivisible. While the Aristote-
lian doctrine held sway for many-centuries yet the notion of atoms
was revived at intervals. Late in the seventeenth century, Boyle
seems to have looked upon chemical combination as the result of
atomic association.

Guided by these early speculations as to the constitution of

FUNDAMENTAL PRINCIPLES 7

matter and influenced by his study of the writings of Sir Isaac
Newton, Dalton seems to have formed a mental picture of the
part played by atoms in the act of chemical combination. After
a few carelessly performed experiments, the results of which ac-
corded with his preconceived ideas, he formulated his atomic
theory.

1 According to this theory matter is composed of extremely mr-
nute, indivisible particles or atoms. Atoms of the same element
are all of equal weight, but atoms of different elements have
weights proportional to their combining numbers. Chemical
compounds are formed by the union of atoms of different kinds.*
This theory offers a simple, rational explanation of the laws of
chemical combination.

Since a chemical compound results from the union of atoms,
each of which has a definite weight, its composition must be in-
variable, — which is the law of definite proportions. Again,
when atoms combine in more than one proportion, for a fixed
weight of atoms of one kind, the weights of the other species of
atoms must bear to each other a simple ratio, since the atoms are
indivisible units. This is clearly the law of multiple proportions.

Finally, the law of combining weights is seen to follow as a
necessary consequence of the atomic theory, since the experimen-
tally determined combining weights bear a simple relation to the
relative weights of the atoms.

At the time when Dalton. proposed his atomic theory, the
number of facts to be explained was comparatively small, but
with the enormous growth of the science of chemistry during the
past century and with the vast accumulation of data, the theory
has proved capable of affording adequate representation of all of
the facts, and has opened the way to many important generaliza-
tions.

While the atomic theory has played a very important part in the
development of modern chemistry, and while we recognize that it
helps to clarify our thinking and enables us to construct a mental
image of tiny spheres uniting to form a chemical compound, yet we
must not forget the fact that these atoms are purely hypothetical.

Faraday has said: "Whether matter be atomic or not, this

8 THEORETICAL CHEMISTRY

much is certain, that granting it to be atomic, it would appear
as it now does." Ostwald believes that in the not distant future
the atomic theory will be abandoned and chemists will free them-
selves from the yoke of this hypothesis, relying solely upon the
results of experiment. He says: "It seems as if the adaptabil-
ity of the atomic hypothesis is near exhaustion, and it is well
to realize that, according to the lesson repeatedly taught by the
history of science, such an end is sooner or later inevitable/'

Combining Weights and Atomic Weights. The problem of
determining the relative atomic weights of the elements would at
first sight appear to be a very simple matter. This might appar-
ently be accomplished by selecting one element, say hydrogen,
it being the lightest known element, as the standard; a compound
of hydrogen and another element may then be analyzed and the
amount of the other element in combination with one part by
weight of hydrogen determined. This weight will be its atomic
weight only when the compound contains but one atom of each
element. To determine the relative atomic weight, therefore, we
must know in addition to the chemical equivalent of the element,
the number of atoms present in the compound. For example, the
analysis of water shows it to contain 8 parts by weight of oxygen
to 1 part by weight of hydrogen; the chemical equivalent of
oxygen is, therefore, 8, and if water contained but one atom of
hydrogen the atomic weight of oxygen would be 8. It can be
shown, however,- that water contains two atoms of hydrogen
and one atom of oxygen, therefore, the atomic weight of oxygen
must be 16. It is evident, therefore, that neither the analysis nor
the synthesis of a compound is sufficient to enable us to determine
the number of atoms of an element combined with one atom of
hydrogen. We shall proceed to the consideration of the methods
by which this problem may be solved.

Gay-Lussac's Law of Volumes. Gay-Lussac in 1808, while
studying the densities of gases before and after reaction, announced
the following law: When gases combine they do so in simple ratios
by volume, and the volume of the gaseous product bears a simple
ratio to the volumes of the reacting gases when measured under like
conditions of temperature and pressure. Thus, one volume of hydro-

FUNDAMENTAL PRINCIPLES 9

gen combines with one volume of chlorine to form two volumes
of hydrochloric acid; one volume of oxygen combines with two
volumes of hydrogen to form two volumes of water (vapor) ; and
one volume of nitrogen combines with three volumes of hydrogen
to form two volumes of ammonia.

In a previous investigation, Gay-Lussac had shown that all gases
behave identically when subjected to changes of temperature and
pressure. This fact, taken together with the simple volumetric
relation just enunciated and the atomic theory, suggested a possible
relation between the number of ultimate particles in equal vol-
umes of different gases.

Berzelius attempted to show that under corresponding condi-
tions of temperature and pressure, equal volumes of different
gases contain the same number of atoms, but he was compelled
to abandon the assumption as untenable.

Avogadro's Hypothesis. It remained for the Italian physicist,
Avogadro,* in 1811, to point out the distinction between atoms
and molecules, terms which had been used almost synonymously
up to his time. He defined the atom as the smallest particle
which can enter into chemical combination, whereas the molecule
is the smallest portion of matter which can exist in a free state.
He then formulated the following hypothesis :f Under the same
conditions of temperature and pressure, equal volumes of all gases
contain the same number of molecules. This hypothesis has been
subjected to such rigid experimental and mathematical tests that
its validity cannot be questioned.

Avogadro's Hypothesis and Molecular Weights. According
to Gay-Lussac when hydrogen and chlorine combine to form hydro-
chloric acid, one volume of hydrogen unites with one volume of
chlorine yielding two volumes of hydrochloric acid.

According to the hypothesis of Avogadro, the number of mole-
cules of hydrochloric acid is double the number of molecules of
hydrogen or of chlorine, and, consequently, each molecule of the
reacting gases must contain at least two atoms. If we take
hydrogen as the unit of our system of atomic weights, its molec-

* Jour, de Phys., 73, 58 (1811).

f Ampere advanced nearly the same hypothesis in 1814.

10 THEORETICAL CHEMISTRY

ular weight must be 2. It is convenient to express molecular
and atomic weights in terms of the same unit, for then the molec-
ular weight of a substance will be simply the sum of the weights
of the atoms contained in the molecule. The determination of
the approximate molecular weight of a substance, therefore, re-
solves itself into ascertaining the mass of its vapor in grams which,
under the same conditions of temperature and pressure, will
occupy the same volume as 2 grams of hydrogen.

This weight is called the gram-mokcular weight or the molar
weight of the substance, while the corresponding volume is known
as the gram-mokcular or molar volume* It is nearly the same for
all gases and at 0° and 760 mm. it may be taken equal to 22.4
liters. The molecular weights obtained from vapor density meas-
urements are approximate only, because of the failure of most
gases and vapors to obey the simple gas laws, a condition essen-
tial to the strict applicability of Avogadro's hypothesis.

Atomic Weights from Molecular Weights. While vapor
density determinations as ordinarily carried out do not give exact
molecular weights, it is an easy matter to arrive at the true values
when we take into consideration the results of chemical analysis.
It is apparent that the true molecular weight must be the sum of
the weights of the constituent elements, these weights being exact
multiples or submultiples of their combining proportions, which
proportions have been determined by analysis alone. We select,
as the true molecular weight, the value which is nearest to the
approximate molecular weight calculated from the vapor density
of the substance. For example, the molecular weight of ammonia,
as computed from its vapor density, is 17.5 or, in other words,
17.5 grams of ammonia occupy the same volume as 2 grams of
hydrogen, measured under the same conditions of temperature
and pressure. The analysis of ammonia shows us that for every
gram of hydrogen, there are present 4.67 grams of nitrogen.
Hence the true molecular weight must contain a multiple of 1 gram
of hydrogen and the same multiple of 4.67 grams of nitrogen.
The problem is, to find what integral value must be assigned to
x In the expression, x (1 -f 4.67), in order that it may give the
closest approximation to 17.5. Clearly if x = 3 the value of the

FUNDAMENTAL PRINCIPLES

expression becomes 17, and this we take to be the true molecular
weight. This gives 3 X 4.67 « 14 as the probable atomic weight
of nitrogen. To decide whether the atomic weight of nitrogen is
a multiple or a submultiple of 14, we must determine the molecu-
lar weights of a large number of gaseous or vaporizable compounds
of nitrogen and select as the atomic weight the smallest quantity
of the element which is present in any one of them.

The following table gives a list of seven gaseous compounds of
nitrogen together with their gram-molecular weights, and the
number of grams of the element in the gram-molecule.

'Compound.

Gram-mol.

Wt.

Grams Nitro-
gen.

Ammonia

Nitric oxide . . . ».

Nitrogen peroxide

Methyl nitrate ,

Cyanogen chloride

61.5

Nitrous oxide. ...

Cyanogen

- It will be observed that the least weight of nitrogen entering
into a gram-molecular weight of any of these compounds is 14
grams, and, therefore, we accept this value as the atomic weight
of the element, although there is still a very slight chance that
in some other compound of nitrogen a smaller weight of the ele-
ment may be found. We shall proceed to point out that there
are methods by which the probable values of the atomic weights
may be checked.

Specific Heat and Atomic Weight. In 1819 the French chem-
ists, Dulong and Petit,* pointed out a very simple relation between
the specific heats of the elements in the solid state and their
atomic weights. This relation, known as the law of Dulong and
Petit, is as follows: The product of the specific heat and the atomic
weight of the solid elements is constant. The value of this constant,
called the atomic heat, is approximately 6.4 A little reflection
will show that an alternative statement of this law is that the
* Ann. Chim. Phys., 10, 395 (1819).

THEORETICAL CHEMISTRY

atoms of the elements in the solid state have the same thermal capac-
ity. The specific heats, atomic weights and atomic heats of
several elements are given in the subjoined table.

Element.

At. Wt.

Sp. Ht.

At. Ht.

Lithium

0 940

6 6

Glucinum . . .

0 410

3 7

Boron (amorphous)
Carbon (diamond) . .

11
12

0 250
0 140

2 8
1 7

Sodium

0 290

6 7

Silicon (crystalline) ...
Potassium

28
39

0 160
0 166

4 5
6 5

Calcium

0 170

6 8

Iron

0 112

6 3

CoDDer

0 093

5 9

Zinc . . . .

0 093

6 1

Silver ... .

108

0 056

6 0

Tin

119

0 054

6 5

Gold

197

0 032

6 3

Mercury

200

0 032

6 4

It is truly remarkable that elements differing as greatly as
lithium and mercury differ, not only in atomic weight but in
other properties as well, should have identical atomic heats. It
will be observed that the atomic heats of boron, silicon, carbon
and glucinum are too low. This departure from the law of Dulong
and Petit is more apparent than real, for in the statement of the
law there is no specification as to the temperature at which the
specific heat should be determined. The specific heats of all
solids vary with the temperature, this variation being greater in
the case of some elements than in that of others. It has been
shown that the specific heats of the above four elements increase
rapidly with rise of temperature and approach limiting values.
As these values are approached the product of specific heat and
atomic weight approximates more and more closely to the mean
value of the constant, 6.4.

The following table gives the values obtained by Weber * for
carbon and silicon.

* Pogg. Ann., 154, 367 (1875).

FUNDAMENTAL PRINCIPLES

CARBON (DIAMOND).

Temperature,
degrees.

Sp. Ht.

At. Ht.

-50

0 0635

0 76

+10

0 1128

1 35

0 1765

2 12

206

0 2733

3 28

607

0 4408

5.30

806

0 4489

5 40

985

0 4589

5 50

CARBON (GRAPHITE).

Temperature,
degrees.

Sp. Ht.

At. Ht.

-50

0.1138

1 37

+ 10

0 1604

1 93

0 1990

2 39

202

0 2966

3 56

642

0 4454

5.35

822

0 4539

5 45

978

0 4670

5 50

SILICON.

Temperature,
degrees

Sp. Ht.

At. Ht

-40

0 136

3 81

+57

0 183

5 13

129

0 196

5.50

232

0 203

5 63

It is evident that this empirical relation can be used to deter-
mine the approximate atomic weight of an element when its

specific heat is known, thus

6.4

atomic weight = ^~~r — , •

specific heat

The law of Dulong and Petit has been of great service in fixing
and checking atomic weights.

About twenty years after the law of Dulong and Petit was
formulated, Neumann * showed that a similar relation holds for
* Pogg. Ann., 23, 1 (1831).

THEORETICAL CHEMISTRY

compounds of the same general chemical character. Neumann's
law may be stated thus: Similarly constituted compounds in the
solid state have the same mokcular heat. Subsequently Kopp *
pointed out that the thermal capacity of the atoms is not appreciably
altered when they enter into chemical combination, or in other words,
the molecular heat of solid compounds is an additive property,
being made up of the atomic heats of the constituent elements.

For example, the specific heat of PbBr2 is 0.054 and its molec-
ular weight is 366.8, therefore, the molecular heat is 0.054 X 366.8
= 19.9. Since there are three atoms in the molecule, 19.9 •*- 3
= 6,6 is their average atomic heat, a value in excellent agree-
ment with the constant in the law of Dulong and Petit. Neu-
mann's law may be used to estimate the atomic heats of elements
which cannot be readily investigated in the solid state. The
following table gives a list of atomic heats of elements in the solid
state derived by means of Neumann's law.

Element.

At. lit.

Element.

At. Ht.

Hydrogen . . .

s
2 3

Carbon

1.8

Oxygen

4 0

Silicon

4,0

Fluorine

5 0

Phosphorus .

5.4

Nitrogen

5.5

Sulphur

5.4

Isomorphism. From a study of the corresponding salts of
phosphoric and arsenic acids, Mitscherlich f observed that they
crystallize with the same number of molecules of water and are
nearly identical in crystalline form, it being possible to obtain
mixed crystals from solutions containing both salts. This sug-
gested to Mitscherlich a line of investigation which resulted, in
1820, in the establishment of the law of isomorphism which bears
his name.

This law may be stated as follows: An equal number of atoms
combined in the same manner yield the same crystal form9 which is
independent of the chemical nature of the atoms and dependent upon
their number and position. Thus, when one element replaces

* Lieb. Ann. (1864), Suppl., 3, 5.

f Ann. Chim. Phys. (2), 14, 172 (1820).

FUNDAMENTAL PRINCIPLES

another in a compound without changing its crystalline form,
Mitscherlich assumed that one element has displaced the other,
atom for atom. For example, having two isomorphous substances,
such as BaCl2.2 H20 and BaBr2.2 H20, we assume that the brom-
ine in the second compound has replaced the chlorine in the first
and, if the atomic weights of all of the elements in the first com-
pound are known, then it is evident that the atomic weight of the
bromine in the second compound can be easily calculated. This
method was largely used by Berzelius in fixing atomic weights and
in checking [the values obtained by the volumetric method. It
should be remembered that the converse of the law of isomorphism
does not hold, since elements may replace each other, atom for
atom, without preserving the same form of crystallization. Many
exceptions to the law have been pointed out. For example,
Mitscherlich himself showed that Na^SC^ and BaMn208 are iso-
morphous and yet the two molecules do not contain the same
number of atoms. Furthermore, careful measurements of the
interfacial angles of crystals have revealed the fact that sub-
stances which have been regarded as isomorphous are only approx-
imately so, thus the interfacial angles of the apparently isomorph-
ous crystalline salts given in the following table differ appreciably.

Salt.

Interfacial Angle.

MgSC
ZnSO
NiSO

)4.7 H2O

4.7 H2O
4.7 H20

89° 26'
88° 53'
88° 56'

Ostwald has suggested that the term homeomorphous be applied to
designate substances which have nearly identical form. At best
the principle of isomorphism is only an approximation and should
be employed with caution.

-Valence. During the latter half of the nineteenth century
the usefulness of the atomic theory was greatly enhanced by the
introduction of certain assumptions concerning the combining
power of the atoms. These assumptions, constituting the so-
called doctrine of valence, were forced upon chemists in order
that a satisfactory explanation might be offered of the phenomenon

16 THEORETICAL CHEMISTRY

of isomerism. A consideration of the following formulas, —
HC1, H20, NH3, CH4, — shows that the power to combine with
hydrogen increases regularly from chlorine, which combines with
hydrogen, atom for atom, to carbon, one atom of which is capable
of combining with four atoms of hydrogen. Either hydrogen or
chlorine, each of which is capable of combining with but one
atom of the other, may be taken as an example of the simplest
kind of atom. Any element like hydrogen or chlorine is called
a univalent element, whereas elements similar to oxygen, nitrogen
and carbon, which are capable of combining with two, three or
four atoms of hydrogen, are called bivalent, trivalent and quadri-
valent elements respectively. Most elements belong to one or
the other of these four classes, although quinquivalent, sexivalent
and septivalent elements are known. The familiar bonds or link-
ages of structural formulas are graphic representations of the
valence of the atoms constituting the molecule. This useful con-
ception of valence has made possible the prediction of the prop-
erties of many compounds before they have been discovered in
nature or in the laboratory.

Atomic Weights. Among the first to recognize the importance
of Dalton's atomic theory was the Swedish chemist, Berzelius.
He foresaw the importance for chemists of a table of exact atomic
weights and in 1810 he undertook the task of determining the
combining weights of most of the known elements. For nearly
six years he was engaged in determining the exact composition of
a large number of compounds and calculating the combining
weights of their constituent elements, thus compiling the first
table of atomic weights.

Numerous investigators since Berzelius have been engaged in
this important work, among whom should be mentioned Stas,
Marignac, Morley and Richards. On two occasions special stimu-
lus was given to such investigations. The first occasion was.in
1815 when Prout suggested that the atomic weights of the elements
are exact multiples of the atomic weight of hydrogen. The values
obtained by Berzelius were incompatible with the hypothesis of
Prout, although the atomic weights of several of the elements
differed but little from integral values. To test the accuracy of

FUNDAMENTAL PRINCIPLES 19

atomic weight of oxygen is taken as 16, and the unit to which all
atomic weights are referred is one-sixteenth of this weight. The
atomic weight of hydrogen on this basis is 1.008. Aside from the
fact that most of the elements form compounds with oxygen which
are suitable for analysis, the atomic weights of more of the ele-
ments approximate to integral values when oxygen instead of hy-
drogen is used as the standard.

The tab'e on page 18 gives the values of the atomic weights
as published by the International Committee on Atomic
Weights for 1917.

CHAPTER II.
CLASSIFICATION OF THE ELEMENTS.

Early Attempts at Classification. Many attempts were made
to classify the elements according to various properties, such as
their acidic or basic characteristics or their valence. In all of
these systems the same elements frequently found a place in more
than one group, and elements bearing little resemblance to each
other were classed together. The early attempts at classifica-
tion based upon the atomic weights of the elements were not
successful owing to the uncertainty as to the exact numerical
values of these constants.

Prout's Hypothesis. In 1815, W. Prout, an English physician,
observed that the atomic weights of the elements, as then given,
did not differ greatly from whole numbers when hydrogen was
taken as the standard. Hence he advanced the hypothesis that
the different elements are* polymers of hydrogen. As has already
been pointed out this hypothesis led Stas to undertake his refined
determinations of the atomic weights of silver, lithium, sodium,
potassium, sulphur, lead, nitrogen and the halogens. As a result
of his investigations he says: "I have arrived at the absolute
conviction, the complete certainty, so far as is possible for a human
being to attain to certainty in such matter, that the law of Prout
is nothing but an illusion, a mere speculation definitely contra-
dicted by experience/' Notwithstanding the fact that Prout's
hypothesis as originally stated was thus disproved by Stas, it still
survived in a modified form given to it by J. B. Dumas, who sug-
gested that one-half of the atomic weight of hydrogen should be
taken as the fundamental unit. When Stas showed that his
experiments excluded this possibility, Dumas suggested that the
fundamental unit be taken as one-quarter of the atomic weight
of hydrogen. Having begun to divide and subdivide, there was no
limit to the process, and the hypothesis fell into disfavor, although

CLASSIFICATION OF THE ELEMENTS

the belief in a primal element, something akin to the protyle
(irpurrj vA.i;) of the ancient philosophers, has survived and in modern
times has reappeared in the electron theory.

Dobereiner's Triads. About 1817 J. W. Dobereiner * observed
that groups of three elements could be selected from the list of the
elements, all of which are chemically similar, and having atomic
weights such that the atomic weight of the middle member is the
arithmetical mean of the first and third members of the group.
These groups of three elements he termed triads. In the follow-
ing table a few of these triads are given.

Element.

At. Wt.

Mean atomic
weight of
triads.

Lithium . . . ...

6 94

Sodium

23 00

I 23.02

Potassium

39 10

Calcium. . . .

40.07

)

Strontium . , .

87 63

> 88.72

Barium

137 37

Chlorine

35.46

)

Bromine . .

79.92

> 80.69

Iodine

126.92

Sulphur

32.07

)

Selenium .

79 2

> 78.78

Tellurium

127.5

Phosphorus ...

31.04

)

Arsenic

74.96

> 75 62

Antimony

120 2

This simple relation, first pointed out by Dobereiner, is clearly
a foreshadowing of the periodic law.

The Helix of de Chancourtois. The idea of arranging the
elements in the order of their atomic weights with a view to
emphasizing the relationship of their chemical and physical prop-
erties, seems to have first suggested itself to M. A. E. B. de Chan-
courtois f in the year 1862, On a right-circular cylinder he traced

* Pogg. Ann., 15, 301 (1825).

f Vis Tellurique, Classement naturel des Corps Simples.

THEORETICAL CHEMISTRY

what he termed a " telluric helix" at a constant angle of 45° to the
axis. On this curve he laid off lengths corresponding to the
atomic weights of the elements, taking as a unit of measure a
length equal to one-sixteenth of a complete revolution of the
cylinder. He then called attention to the fact that elements with
analogous properties fall on vertical lines parallel to the generatrix.
Being a mathematician and a geologist he did not express himself
in such terms as would attract the attention of chemists and con-
sequently his work remained unnoticed until recent times.

The Law of Octaves. In 1864 J. A. R. Newlands* pointed
out that if the elements are arranged in the order of their atomic
weights, the eighth element has properties very similar to the
first; the ninth to the second; the tenth to the third; and so on,
or to employ Newlands' own words: "The eighth element starting
from a given one is a kind of repetition of the first, like the eighth note of
an octave in music." This peculiar relationship, termed by New-
lands the law of octaves, is brought out in the following table.

Notwithstanding the fact that its author was ridiculed and his
paper returned to him as unworthy of publication in the proceed-
ings of the Chemical Society, this generalization must be regarded
as the immediate forerunner of the periodic law.

The Periodic Law. Quite independently of each other and
apparently in ignorance of the work of Newlands and de Chan-
courtois, Mendel6eff f in Russia and Lothar Meyer in Germany,
gained a far deeper insight into the relations existing between
the properties of the elements and their atomic weights. In 1869
Mendeteeff wrote: — "When I arranged the elements according
to the magnitude of their atomic weights, beginning with the
smallest, it became evident that there exists a kind of periodicity

* Chem. News, 10, 94 (1864), Ibid., 12, 83 (1865).
t Lieb. Ann. Suppl., 8, 133 (1874).

CLASSIFICATION OF THE ELEMENTS 23

in their properties. I designate by the name 'periodic law' the
mutual relations between the properties of the elements and their
atomic weights; these relations are applicable to all the elements
and have the nature of a periodic function/' This important
generalization may be briefly stated thus: The properties of the
elements are periodic functions of their atomic weights.

The original table of Mendeleeff has been amended and modified
as new data has accumulated and new elements have been dis-
covered. The accompanying table, though containing several
new elements and an entirely new group, is essentially the same as
that of Mendeleeff. It consists of nine vertical columns, called
groups, and twelve horizontal rows termed series or periods. The
second and third periods contain eight elements each, and are
known as short periods, while in the fourth series, starting with
argon, it is necessary to pass over eighteen elements before another
element, krypton, is encountered which bears a close resemblance
to argon: such a series of nineteen elements is called a long period.
The entire table is composed of two short and five long periods,
the last one being incomplete. The positions of the elements are
largely determined by their chemical similarity to those in the
same group, the hyphens indicating the positions of undiscovered
elements. The elements in Group VIII, presented difficulties
when Mendel6eff attempted to place them according to their
atomic weights and so he was obliged to group them by themselves.
This group has wittily been designated as " the hospital for incur-
ables. " An examination of the table shows that the valence of
the elements toward oxygen progresses regularly from Group O,
containing elements which exhibit no combining power, up to
Group VIII, where it attains a maximum value of eight in the
case of osmium. The valence toward hydrogen on the other hand
increases regularly from Group VII to Group IV in which the
elements are quadrivalent.

THEORETICAL CHEMISTRY

2ffl
Off?

1 S

Stfl

is
M

r-<

000

c59S

i-H ^1 *

°^ °

ZH W

O S

,-•!

Z! u2 v® «

pn pn y^ ^^

00 OS

"8 "S "g

"8 "g "
Jl |1 |

M t-5 (-3

CIASSIFICATION OF THE ELEMENTS 25

The formulas of the typical oxides and hydrides of the elements
in the several groups are indicated at the top of each vertical
column in the table, where R denotes any element in the group.
The valence of elements in the long periods arc apt to be variable.
The elements in the second series are frequently called bridge
elements, since they bear a closer relation to the elements in the
next adjacent group than they do to any other members of the
same group in succeeding series. The members of the third series
are styled typical elements, because they exhibit the general prop-
erties and characteristics of the group. Each group is divided into
subgroups, the elements on the right and left sides of a column
forming families, the members of which are more closely related
than are all of the elements included within the group. In other
words we detect a kind of periodicity within each group.

In any given series the element with lowest atomic weight
possesses the strongest basic character. Thus we find the strongly
basic, alkali metals on the left side of the table, while on the right
side are the acidic elements such as the halogens and sulphur.
In fact, the strictly non-metallic elements are confined to the
upper right-hand corner of the table.

Similarly, as we pass from the top to the bottom of the table,
we observe a progressive change in the base-forming tendency of
the elements; i.e., as the atomic weight increases, the metallic
character of the elements in each group becomes more pronounced.

Periodicity of Physical Properties. Lothar Meyer, as has been
pointed out, discovered the periodic relations of the elements at
about the same time as Mendel^eff . His table differed but slightly
from that already given. The most important part of Meyer's *
work, however, was in pointing out that various physical proper-
ties of the elements are periodic functions of their atomic weights
We know today that such properties as specific gravity, atomic
volume, melting point, hardness, ductility, compressibility, ther-
mal conductivity, coefficient of expansion, specific refraction, and
electrical conductivity are all periodic. When the numerical
values of these properties are plotted as ordinates against their
atomic weights as abscissae, we obtain wave-like curves similar to
* Die Modernen Theorien der Chemie.

THEORETICAL CHEMISTRY

those shown in Fig. 2. The specific heats of the elements are an
exception to the general rule. According to the law of Dulong and
Petit, the product of specific heat and atomic weight is a constant,
and consequently the graphic representation of this relation must
be an equilateral hyperbola.

Applications of the Periodic Law. Mendel£eff pointed out the
four following ways in which the periodic law could be employed:
— (1) The classification of the elements; (2) The estimation of the

A/I/

Fig. 2.

atomic weights of elements; (3) The prediction of the properties
of undiscovered elements; and (4) The correction of atomic
weights.

1. Classification of Elements. The use of the periodic law in
this direction has already been indicated. It is without doubt
the best system of classification known and is to be ranked among
the great generalizations of the science of chemistry.

2. Estimation of Atomic Weights. Because of experimental
difficulties it is not always possible to fix the atomic weight of an
element by determinations of the vapor densities of some of its
compounds, or by a determination of its specific heat. In such
cases the .periodic law has proved of great value. An historic

CLASSIFICATION OF THE ELEMENTS 27

example is that of indium, the equivalent weight of which was
found by Winkler to be 37.8. The atomic weight of the element
was thought to be twice the equivalent weight or 75.6. If this
were the correct value it would find a place in the periodic table
between arsenic and selenium. Clearly there is no vacancy in
the table at this point and furthermore its properties are not
allied to those of arsenic or selenium. Mendel<5eff proposed to
assign to it an atomic weight three times its equivalent weight or
113.4, when it would fall between cadmium and tin in the table.
This would bring it in the same group with aluminium, the typical
element of the group, to which it bears a close resemblance. This
suggestion of Mendeteeff s was confirmed by a subsequent deter-
mination of the specific heat of indium.

3. Prediction of Properties of Undiscovered Elements. At the
time when Mendeleeff published his first table there were many
more vacant spaces than exist in the present periodic table. He
ventured to predict the properties of many of these unknown
elements by means of the average properties of the two neighbor-
ing elements in the same series, and the two neighboring elements
in the same subgroup. These four elements he termed atomic
analogues. The undiscovered elements Mendel6eff designated by
prefixing the Sanskrit numerals, eka (one), dwi (two), tri (three),
and so on, to the names of the next lower elements of the sub-
group. When the first periodic table was published there were
two vacancies in Group III, the missing elements being called by
Mendeleeff eka-alurninium and eka-boron, while in Group IV
there was a vacancy below titanium, the missing element being
called eka-silicon. The subsequent discovery of gallium, scandium
and germanium, with properties nearly identical with those pre-
dicted for the above hypothetical elements, served to strengthen
the faith of chemists in the periodic law. The following table
illustrates the accuracy of Mendel6efFs prognostications: in it is
given a comparison of a few of the properties of the hypothetical
element, eka-silicon, as predicted by Mendeleeff in 1871, and tbf*
corresponding observed properties of germanium, discovered by
Winkler fifteen years later.

THEORETICAL CHEMISTRY

Eka-sihcon, Es.

Germanium, Ge.

Atomic weight, 72

Specific gravity, 5.5.

Atomic volume, 13.

Metal dirty gray, and on ignition

yields a white oxide, EsO2.
Element decomposes steam with

difficulty.
Acids have slight action, alkalies no

pronounced action.

Action of Na on EsOz or on EsK2F6
gives metal.

The oxide EsO2 refractory.

Specific gravity of EsO2, 4.7.

Basic properties of EsO2 less marked
than Ti(>2 and SnO2, but greater
than SiO2.

Forms hydroxide soluble in acids,
and the solutions readily decom-
pose forming a metahydrate.

EsCl4 a liquid with a b.p. below 100°
and a sp. gr. of 1.9 at 0°.

EsF4 not gaseous.

Es forms a compound Es(C2H5)4 boil-
ing at 160°, and with a sp. gr. 0.96.

Atomic weight, 72.3.
Specific gravity, 5.47.
Atomic volume, 13.2.
Metal grayish- white, and on igni-
tion yields a white oxide, GeO2.
Element does not decompose water.

Metal not attacked by HC1, but
acted upon by aqua regia.

Solutions of KOH have no action.
Oxidized by fused KOH.

Ge obtained by reduction of
with C, or of GeK2F6 with Na.

The oxide GeOa refractory.

Specific gravity of GeO2, 4.703.

Basic properties of GeO2 feeble.

Acids do not ppt. the hydroxide
from dii. alkaline solutions," but
from cone, solutions, acids ppt.
GeO or a metahydrate.

GeCl4 a liquid with a b.p. of 86°,
and a sp. gr. at 18° of 1.887.

GeF4.3 H2O a white solid.
Ge forms a compound Ge(C2H6)4
boiling at 160° and with a sp. gr.
slightly less than water.

4. Correction of Atomic Weights. When an element falls in a
position in the periodic table where it clearly does not belong,
suspicion as to the correctness of its atomic weight is immediately
aroused. Frequently a redetermination of the atomic weight has
revealed an error which, when corrected, has resulted in assigning
the element to a place among its analogues. Formerly the
accepted atomic weights of osmium, iridium, platinum and gold
were in the order

Os > Ir > Pt > Au.

But from analogies existing between osmium, ruthenium and iron
and the disposition of the preceding members of Group VIII,
Mendel^eff predicted that the atomic weights were in error and
that the order of the elements should be

Os < Ir < Pt < Au.

CLASSIFICATION OF THE ELEMENTS 29

Subsequent atomic weight determinations by Seubert substantiated
Mendel6eff's prediction.

Defects in the Periodic Law. While the arrangement of the
elements in the periodic table is on the whole very satisfactory,
there are several serious defects in the system which should be
pointed out. At the very outset there is difficulty in finding a
place for hydrogen in the system. The element is univalent and
falls either in Group I, with the alkali metals, or in Group VII with
the halogens. While the element is electro-positive it cannot be
considered to possess metallic properties. It forms hydrides with
some of the metallic elements and can be displaced by the halo-
gens from organic compounds. These facts make it extremely
difficult to decide whether hydrogen should be placed in Group I
or Group VII. The idea has been advanced that hydrogen is
the only known member of the first series of the periodic table.

These hypothetical elements have been styled proto-ekments,
the successive members of the series being, proto-glucinum, proto-
boron and so on to the last element in the series, proto-fluorine. To
find a suitable location for the rare-earth elements in the periodic
system is another difficulty which has not been satisfactorily met.
Brauner considers that these elements should all be grouped
together with cerium (at. wt. = 140.25), but owing to our limited
knowledge of the properties of these elements it seems better to
defer attempting to place them for the present. In the group of
non-valent elements the atomic weight of argon is distinctly higher
than that of potassium in the next group. There can be little
doubt that the values of the atomic weights are correct and it
is evidently impossible to interchange the positions of these two
elements in the periodic table, since argon is as much the analogue
of the rare gases as potassium is of the alkali metals. A similar
discrepancy occurs with the elements, terrurium and iodine. The
atomic weight of the former element is appreciably higher than
that of the latter and, notwithstanding the attempts of numerous
investigators to prove tellurium to be a complex of two or more
elements, nothing but failure has attended their efforts. Still
another anomaly is encountered in Group VII, where manganese
is classed with the halogen family, to which it bears much less

30 THEORETICAL CHEMISTRY

resemblance than it does to chromium and iron, its two immediate
neighbors.

As has already been mentioned, Group VIII is made up of non-
conformable elements. If the properties of the elements are
dependent upon their atomic weights, it should be impossible for
several elements having almost identical atomic weights and
different properties to exist, and yet such is the case with the
elements of Group VIII. The elements copper, silver and gold,
while not closely resembling the other members of Group VIII,
are much more closely allied to them than to the alkali metals
with which they are also classed. Notwithstanding its imper-
fections, the periodic law must be regarded as a truly wonderful
generalization which future investigations will undoubtedly show
to be but a fragment of a more comprehensive law.

CHAPTER III.
THE ELECTRON THEORY.

Conduction of Electricity through Gases. Within recent
years the discovery of new facts relative to the conduction of
electricity through gases has led to the development of the so-
called electron or corpuscular theory of matter. Under ordinary
conditions gases are practically non-conductors of electricity, but
when a sufficiently great difference of potential is established
between two points within a gas it is no longer able to withstand
the stress, and an electric discharge takes place between the points.
The potential necessary to produce such a discharge is quite high,

>pump
Fig. 3.

several thousand volts being required to produce a spark of one
centimeter length in air at ordinary pressures. The pressure of
the gas has a marked effect upon the character of the discharge
and the potential required to produce it. If we make use of a
glass vessel similar to that shown in Fig. 3, the effect of pressure
on the nature of the discharge may be studied.

This apparatus consists of a straight glass tube about 4 cm. in
diameter and 40 cm. long, into the ends of which platinum elec-
trodes are sealed. To the side of the vessel a small tube is sealed
so that connection may be established with an air-pump and
manometer. If the electrodes are connected with the terminals
of an induction coil and the pressure within the tube be gradually
diminished, the following changes in the character of the dis-

THEORETICAL CHEMISTRY

charge will be observed. At first the spark becomes more uni-
form and then broadens out, assuming a bluish color. When a
pressure of about 0.5 mm. is
reached, the negative electrode
or cathode will appear to be
surrounded by a thin luminous
layer; next to this will be a
dark region, known as the

Crookes' dark space; adjoining „. 4 opump

this will be a luminous portion,

called the negative glow, and beyond this will be seen another dark
region which is frequently referred to as the Faraday dark space.
Between the Faraday dark space and the positive electrode or
anode is a luminous portion, called the positive column. By a slight
variation of the current and pressure the positive column can be
caused to break up into alternate light and dark spaces or strice,
the appearance of which is dependent upon various factors such as

Fig. 5.

the nature of the gas and the size of the tube. If we use a modi-
fication of this tube, such as is shown in Fig. 4, and diminish the
pressure to about 0.01 mm., a new phenomenon will be observed.
The positive column will vanish and the walls of the tube opposite
the cathode will become faintly phosphorescent. The color of
the phosphorescence will depend upon the nature of the glass:
if the tube is made of soda glass, the glow will be greenish yellow,
while with lead glass the phosphorescence will be bluish. The
phosphorescence is due to the bombardment of the walls of the

THE ELECTRON THEORY 33

tube by very minute particles projected normally from the cathode.
These streams of particles are called the cathode rays.

Some Properties of Cathode Rays. The following are among
the most important properties of the cathode rays: —

1. The cathode rays travel in straight lines normal to the cathode:
and they cast shadows of opaque objects placed in their path. This
property may be demonstrated by means of the apparatus shown
in Fig. 5, where a small metallic Maltese cross is interposed in the
path of the rays, a distinct shadow being cast on the opposite wall
of the tube. The cross may be hinged at the bottom so that
it can be dropped out of the path of the rays, when the usual
phosphorescence will be obtained.

2, The cathode rays can produce mechanical motion. By means

Fig. 6.

of the apparatus due to Sir William Crookes, Fig. 6, this prop-
erty of the cathode rays may be demonstrated. Within the
vacuum tube is placed a small paddle wheel which rolls horizon-
tally on a pair of glass rails. When the current is applied to the
tube, the wheel will revolve, moving away from the cathode. By
reversing the current, the wheel will stop and then rotate in the
opposite direction owing to the reversal of the direction of the
cathode stream.

3. The cathode rays cause a rise of temperature in objects upon
which they fall. In the tube shown in Fig. 7, the anode consists
of a small piece of platinum: this is placed at the center of curva-
ture of the spherical cathode. After pumping down to the
proper pressure, if a strong discharge be sent through the tube,

THEORETICAL CHEMISTRY

Jr To pump

the anode will begin to glow, and if the action of the current be
continued long enough, the platinum plate may be rendered in-
candescent, thus showing the
marked heating effect of the
cathode rays.

4. Many substances become
phosphorescent on exposure to
the cathode rays. If the cathode

rays be directed upon different p. 7

substances, such as calc-spar,

barium platino-cyanide, willemite, scheelite and various kinds
of glass, beautiful phosphorescent effects may be observed. This
phosphorescent property is useful in observing and experimenting
with the cathode rays.

5. The cathode rays can be deflected from their rectilinear path by
a magnetic field. In studying the magnetic deviation of the
cathode rays a tube similar to that shown in Fig. 8 has been
found very satisfactory. An aluminium diaphragm, A, pierced

IJL

Fig. 8.

with a 1 mm. hole, is placed in front of the cathode while at the
opposite end of tube is placed a phosphorescent screen, D. When
the discharge takes place a circular phosphorescent spot will
appear on D. If the tube be placed between the poles of an
electromagnet, the phosphorescent spot will move at right angles
to the direction of the magnetic field. On reversing the polarity
of the magnet the spot will move in the opposite direction.
Furthermore the direction of the deflection will be found to be

THE ELECTRON THEORY

similar to that produced by a negative charge of electricity mov-
ing in the same direction as the cathode ray.

6. The cathode rays can be deflected from their rectilinear path
by an electrostatic field. The same tube which was used in observ-
ing the magnetic deflection may be employed in studying the
effect of an electrostatic field. Two insulated metal plates, B
and C, are placed on opposite sides of the tube and parallel to
each other. When the tube is in action, if a difference of poten-
tial of several hundred volts be applied to the plates, the phos-
phorescent spot on D will be found to move, the direction of the

Elect.1

Fig. 9.

motion being the same as that of a negatively charged body under
the influence of an electrostatic field. Reversal of the field causes
the phosphorescent spot to move to the opposite side of the
screen.

7. The cathode rays carry a negative charge. Probably the most
important charactew|tic of the cathode rays is their ability to
carry a negative charge. While the magnetic and electrostatic
deviation of the rays made this fact more than probable, it re-
mained for Perrin to demonstrate that a negative electrification
accompanies the cathode stream. A modification of Perrin's
apparatus due to J. J. Thomson is shown in Fig. 9. It con-

36 THEORETICAL CHEMISTRY

sists of a spherical bulb to which is sealed a smaller bulb and a long
side tube. The small bulb contains the cathode C and the anode
A. The anode consists of a tight-fitting brass plug pierced by a
central hole of small diameter. The side tube, which is out of
the direct range of the cathode rays, contains two coaxial metallic
cylinders insulated from each other, each being perforated with a
narrow transverse slit: D is earth-connected and B is connected
with an electrometer by means of the rod F. When the tube has
been pumped down to the proper pressure for the production of
cathode rays, a phosphorescent spot will appear at E directly
opposite the cathode C. Upon testing B for possible electrifica-
tion by means of the electrometer, it will be found to be uncharged.
If the cathode stream be deflected by means of a magnet so that
the rays fall upon J5, a sudden charging of the electrometer will be
observed, proving that B is becoming electrified. Upon deflect-
ing the rays still further so that they are no longer incident upon
By the accumulation of charge will immediately cease. If the
electrometer be tested for polarity, it will be found to be negatively
charged, thus proving the charge carried by the cathode rays to
be negative.

8. The cathode rays can penetrate thin sheets of metal. In 1894
Lenard constructed a vacuum tube fitted with an aluminium
window opposite the cathode. He showed that the cathode rays
passed through the aluminium and are absorbed by different sub-
stances outside of the tube, the absorption varying directly with
the density of the substance.

9. The cathode rays when directed into moist air cause the forma-
tion of fog. This phenomenon has been shown by C. T. R. Wilson
to be due to the minute particles in the cathode stream acting as
nuclei upon which the water vapor can condense.

Velocity of the Cathode Particle. Since the cathode rays
consist of minute, negatively-charged particles which can be
deflected by a magnetic and an electrostatic field, it is possible
to measure their speed and to compute the ratio of the mass of a
particle to its charge. The special form of tube shown in Fig.
10 was devised for the purpose by J. J. Thomson. It consists
of a glass tube about 60 cm. in length, furnished with a flat cir-

THE ELECTRON THEORY

cular cathode, C, and an anode, A, in the form of a cylindrical
brass plug about 2.5 era. in length, pierced by a central hole 1 mm.
in diameter. Another brass plug, B, is placed about 5 cm. away
from A-, the two holes being in exactly the same straight line, so
that a very narrow bundle of rays may pass along the axis of the
tube and fall upon the phosphorescent screen at the opposite
end of the tube. Upon this screen is a millimeter scale, SS'.
Two parallel plates, D and E, are sealed into the tube for the pur-
pose of establishing an electrostatic field. When the tube is
connected with an induction coil or other source of high-potential,
a phosphorescent spot will appear at F. If a strong magnetic
field be applied, the lines of force being at right angles to the plane

Fig. 10.

of the diagram, the rays will be deflected vertically and the spot
on the screen will move from F to G.

Let H denote the strength of the magnetic field and let m, e
and v represent respectively the mass, charge and velocity of a
cathode particle. A magnetic field, H, acting at right angles to
the line of flight of the cathode particle will exert a force,
Hev, which will tend to deflect the particle from a rectilinear
path. This force must be equal to the centrifugal force of the
moving particle ^pttng outwards along its radius of curvature
Therefore

Hev = — >

„

Hr = — •

38 THEORETICAL CHEMISTRY

Since H and r can both be measured, the ratio, — , can be deter-

mined. Now if a difference of potential be established between
D and E, and the lines of force in the electrostatic field have the
proper direction, it will be possible to alter the strength of the field
so as to just counterbalance the effect of the magnetic field, and
bring the phosphorescent spot back to F again. Under these
conditions, if X denotes the strength of the electrostatic field, we
have

Xe = Hev,
or

*-f- (2)

Since X and H can both be measured, v can be calculated, and by
introducing the value so obtained into equation (1), the ratio e/m
can be evaluated. By this method the average value of v has
been found to be 2.8 X 109 cm. per second, while 1.7 X 107 is
the mean value of a large number of determinations of the ratio
e/m.

Comparison of the Ratio of Charge to Mass for the Cathode
Particle with that for the Ion in Electrolysis. The ratio of the
charge carried by an ion in electrolysis to its mass can be easily
computed. Thus it may be shown that the ratio of the charge E,
of the hydrogen ion to its mass, M , in electrolysis is about 1 X 104
C.G.S. units or

E
-z = I X 104 approximately.

The mass of the hydrogen ion may be considered to be identical
with that of the hydrogen atom, the lightest atom known. Com-
paring the value of e/m for the cathode particle with the value of
E/M for the hydrogen ion in electrolysis, it is evident that the
former is about 1700 times greater than the latter.

Charge Carried by the Cathode Particle. Until the value
of the charge carried by the cathode particle has been determined,
it is clearly impossible to compute its mass. Thus, if we consider

THE ELECTRON THEORY 39

the last statement of the preceding paragraph, which may be
formulated as follows: —

e/m : E/M :: 1700 : 1,

the proportion will remain unaltered whether m = Af/1700 and
e = E, or e = 1700 and m = M. The method employed to deter-
mine the charge carried by a cathode particle is too complicated
for a detailed description in this place; merely the general outline
will be given. Upon suddenly expanding a volume of saturated
water vapor, its temperature is lowered, and a cloud forms, each
particle of dust present serving as a nucleus for a fog particle. If
sufficient time be allowed for the mist to settle and the vapor to
become saturated again, a repetition of the preceding process" will
result in the formation of less mist, owing to the presence of fewer
dust particles. By repeating the operation enough times the
space may be rendered dust free. As has already been pointed
out, cathode particles serve as nuclei for the condensation of
water vapor, their function being similar to that of dust particles.
It has been shown by Sir George G. Stokes that if a drop of water
of radius r, be allowed to fall through a gas of viscosity 17, then the
velocity with which the drop falls is given by the equation

.-?.* (3,

where g is the acceleration due to gravity. The viscosity of air
at any temperature being known, a cloud can be produced in an
appropriate chamber by expansion of water vapor in the presence
of cathode particles and the speed, y, with which the cloud falls
can be measured, and hence r can be calculated by means of equa-
tion (3). If m is the total mass of the cloud and n is the number of
drops per cubic centimeter, then

m = 4/3 mrr3 (density of water = 1).

From a simple application of thermodynamics m may be deter-
mined. Knowing the values of m and r, the number of drops
in the cloud, n, which is the same as the number of cathode par-
ticles, can be calculated. It is a simple matter to measure the
total charge in the expansion chamber, and dividing this by the
total number of charged particles, gives the charge carried by a

40 THEORETICAL CHEMISTRY

single particle. The latest determinations of J. J. Thomson show
this to be 3,4 X 10~10 electrostatic unit. This is practically
identical with the calculated value of the charge on the hydro-
gen ion in electrolysis, or e = E and therefore m = M/1700; the
mass of the cathode particle is 1/1700 of the mass of the atom
of hydrogen. The cathode particle has the smallest mass yet
known and has been called the corpuscle or electron.

An ingenious modification of the foregoing method devised by
Millikan,* has made it possible to determine e with extreme
accuracy. Millikan gives as the mean of a large number of deter-
minations of e, the following value which he states is in error by
less than 0.1 per cent:

e = 4.4775 X 10~10 electrostatic units.

For purposes of comparison, the following values of e obtained by
other investigators using different experimental methods are here
given: —

(a) Making use of available data on radiant energy, Planck
calculated e = 4.69 X lO"10.

(6) By counting the number of scintillations produced by a
known weight of polonium and measuring the total charge, Regener
found e = 4.79 X 1Q-10.

(c) By counting the number of a-particles escaping from a given
amount of radium bromide and measuring the total charge,
Rutherford and Geiger calculated e = 4.65 X 10~10.

The Avogadro Constant. The actual number of molecules in
one gram-molecule or the actual number of atoms in one gram-
atom of a gas is called the Avogadro Constant. The most accurate
method for the calculation of this constant involves the elementary
charge of electricity, e.

The quantity of electricity carried by one gram equivalent in
electrolysis has been found to be 96,500 coulombs (see p. 390).
This quantity, known as the faraday, and commonly designated
by F, bears the same relation to e that the number expressing its
atomic weight bears to the actual weight of an atom. The relation

*Phil. Mag., 6, 19, 209 (1910); Phys. Rev., 39, 349 (1911); Trans. Am.
Electrochem. Soc., 21, 186 (1912).

THE ELECTRON THEORY 41

between the Avogadro constant N and the ionic and electronic
charges F and e, is given by the equation

N = L.

Substituting the above values in this equation and converting
coulombs into electrostatic units, we have '

,r 96500 X 3 X HP _, - 1fm
N = 4.4775 X 10-" = *** X *°*'

Other Sources of Electrons. Electrons may be produced by
other agents than cathode rays. Thus, electrons are emitted by
radium and other radioactive substances, by metals and amalgams
under the influence of ultra-violet light, and also by gas flames
charged with the vapors of salts. It has been shown that from
whatever source an electron is derived, the value of the ratio e/m
remains constant. This interesting fact has led Thomson to sug-
gest that the electron may be regarded as "one of the bricks of
which the atom is built up."

Before entering upon a discussion of modern views of atomic
structure, however, it will be necessary to summarize very briefly
some of the salient facts of radiochemistry.

CHAPTER, IV.
RADIOACTIVITY.

Discovery of Radioactivity. The first radioactive substance
was discovered by Henri Becquerel* in 1896. It had been shown
by Roentgen in the previous year that the bombardment of the
walls of a vacuum tube by the cathode stream, gives rise to a now
type of rays, which, because of their puzzling characteristics, he
called X-rays. The portion of the tube where these rays originate
was observed to fluoresce brilliantly, and it was at once assumed that
this fluorescence might be the cause of the new type of radiation.

Many substances were known to fluoresce under the stimulus of
the sun's rays, and it wa>s natural, in the light of Roentgen's dis-
covery, that all substances which exhibit fluorescence should be
subjected to careful examination. Among those who became inter-
ested in these phenomena was BecquereL He studied the action of
a number of fluorescent substances, among which was the double
sulphate of potassium and uranium. This salt-, after exposure to
sunlight, was found to emit a radiation capable of affecting a care-
fully protected photographic plate. Further investigation proved
that the fluorescence had nothing to do with the photographic
action, since both uranous and uranic salts were found to exert
similar photographic action, notwithstanding the fact that uranous
salts are not fluorescent. The photographic activity of both
uranous and uranic salts was found to be proportional to their
content of uranium. Becquerel also showed that preliminary
stimulation by sunlight was wholly unnecessary. Uranium salts
which had been kept in the dark for years were found to be just as
active as those which had been recently exposed to brilliant sun-
light. The properties of the rays emitted by uranium salts differ
in many respects from those of the X-rays. The rate of emission
of the uranium rays remains unaltered at the highest or the lowest

* Compt. rendus, 122, 420 (1896).
42

RADIOACTIVITY 43

obtainable temperatures. The entire behavior of these salts justi-
fies the conclusion that the continuous emission of penetrating
rays is a specific property of the element uranium itself. This
property of spontaneously emitting radiations capable of penetra-
ting substances opaque to ordinary light is called radioactivity.

Discovery of Radium. Shortly after the discovery of the
radioactivity of uranium, the element thorium and its compounds
were also found to be radioactive. As a result of a systematic
examination by Mme. Curie * of minerals known to contain
uranium or thorium, it was learned that many of these were much
more radioactive than either uranium or thorium alone. Thus,
pitchblende, one of the principal ores of uranium, was found to
be four times more active than uranium alone, and chalcolite, a
phosphate of copper and uranium, was found to be at least twice
as active as uranium. On the other hand, when a specimen of
artificial chalcolite, prepared in the laboratory from pure materials,
was examined, its activity was found to be proportional to the
content of uranium. Mme. Curie concluded from this result that
natural chalcolite and pitchblende must contain a minute amount
of some substance much more active than uranium.

With the assistance of her husband, Mrne. Curie undertook the
task of separating this unknown substance from pitchblende.
Pitchblende is an extremely complex mineral and its systematic
chemical analysis calls for skill and patience of a high order. With-
out entering into details as to the analytical procedure, it must
suffice here to state the results obtained. Associated with bismuth,
a very active substance was discovered, to which Mme. Curie gave
the name polonium in honor of her native land, Poland. In like
manner, an extremely active substance was found associated with
barium in the alkaline earth group. The substance was called
radium because of its great radioactivity.

While the isolation of pure polonium is extremely difficult and,
while sufficient quantities have not been obtained to permit de-
terminations of its physical properties, the isolation of radium
in relatively large amounts is readily accomplished. The pure
bromides of radium and barium are prepared together and the

* Compt. rendus, 126, 1101 (1898).

44 THEORETICAL CHEMISTRY

two salts are then separated by a series of fractional crystalliza-
tions. That the salts of barium and radium are very similar in
chemical properties is shown by the fact that they separate to-
gether from the same solution. The atomic weight of radium has
been determined by several investigators, the accepted value being
226. It is thus, with the exception of uranium, the heaviest known
element.

In 1910, Mme. Curie * succeeded in obtaining metallic radium.
It is a metal possessing a silvery luster, dissolving in water with
energetic evolution of hydrogen and tarnishing rapidly in air with
the formation of the nitride.

It is estimated that one ton of pitchblende contains approxi-
mately 0.2 gram of radium.

Other Radioactive Substances. Shortly after the discovery
of radium and polonium by the Curies, Debierne f discovered
another radioactive element in pitchblende. This element, which
he named actinium, was found associated with the iron group in
the course of the analysis of the mineral.

In 1906, Boltwood f discovered in pitchblende, and in several
other uranium^ minerals, the presence of still another radioelement
which he named ionium. Ionium is much more active than
thorium to which it bears such a close resemblance that the two
elements cannot be separated from each other.

The lead which is obtained from uranium and thorium minerals
is found to be slightly radioactive, the activity being attributed
to the presence of a small proportion of a constituent called radio-
lead, the chemical properties of which resemble those of ordinary
lead. It is interesting to note that recent determinations by
Richards § of the atomic weight of lead obtained from different
sources, reveal differences greater than the possible experimental
errors of the determinations. Thus, the values of the atomic
weight of lead from pitchblende and from thorite were found to
be 206.40 and 208.4 respectively, while the value of the atomic
weight of ordinary lead is 207.15.

* Compt. rendus, 151, 523 (1910).

t Compt. rendus, 129, 593 (1899).

i Am. Jour. ScL, 22, 537 (1906).

§ Jour. Am. Chem. Soc., 36, 1329 (1914).

RADIOACTIVITY 45

About thirty other radioactive substances have been separated
and many of their properties have been determined. AJ1 of these
radio-elements have been shown to be the lineal descendants of one
or the other of the two parent elements, uranium or thorium.

lonization of Gases. The radiations emitted by radioactive
substances have the power of rendering the air through which they
pass conductors of electricity. To account for this action, Thom-
son and Rutherford formulated the theory of gaseous ionization.
According to this theory, which has since been experimentally
confirmed, the radiations break up the components of the gas into
positive and negative carriers of electricity called ions. If two
parallel metal plates are connected to the terminals of a battery
and a radioactive substance is placed between them, the air will be
ionized and, owing to the movement of the positive and negative
ions toward the plates of opposite sign, an electric current will
pass between the plates. If the electric field is weak, the mutual
attraction between the positive and negative ions will cause many
of them to recombine before reaching the plates and the resulting
current will be small. As the strength of the field is increased, the
greater will be the speed of the ions toward the plates and the
smaller will become the tendency to recombination. Ultimately,
with increasing strength of field all of the ions will be swept to the
plates as fast as they are formed and the ionization current will
attain a maximum value. This limiting or saturation current
affords the most accurate method for the measurement of radio-
activity.

The method is so sensitive that, by means of it alone, it is possible
to detect amounts of radioactive products far beyond the reach of
the balance or the spectroscope. The theory of gaseous ionization
has been confirmed in several different ways, but one of the most
striking verifications of the theory is that due to C. T. R. Wilson.
Making use of the fact that the ions tend to condense water vapor
around themselves as nuctei, Wilson succeeded in actually photo-
graphing the path of an ionizing ray in air.

Photographic Action of Radiations. It has already been
pointed out that the radiations from radioactive substances are
capable of affecting a photographic plate. The photographic

46 THEORETICAL CHEMISTRY

action of the radiations has been employed quite extensively in
studying radioactive phenomena from a purely qualitative stand-
point. The method employed, consists in exposing the photo-
graphic plate, which has been previously wrapped in opaque black
paper, to the action of the radiations. The time of exposure
varies with the nature of the substance under examination, a few
minutes being required for highly active preparations while sev-
eral days or even weeks may be needed for preparations of low
activity.

Phosphorescence Induced by Radiations. A screen covered
with crystals of phosphorescent zinc sulphide is rendered luminous
when exposed to fairly intense radiation from a radioactive
substance. This phenomenon has been shown to be due to the
bombardment of the crystals of zinc sulphide by the so-called a-
rays (see below). When the screen is examined with a lens the
phosphorescence is seen to consist of a series of scintillations of
very short duration.

Nature of the Radiations. The ionizing, photographic, and
luminescent properties of the radiations from radioactive sub-
stances are not sufficient to differentiate them from cathode rays
or X-rays, although each of these properties may be employed in
determining their intensity.

Evidence of the composite character of the radiations was fur-
nished by a study of their penetrating power as well as by investi-
gations of the behavior of the rays when subjected to the action of
magnetic and electric fields.

A thin sheet of aluminum or a few centimeters of air was found
sufficient to cut off a large percentage of the rays.

The unabsorbed portion of the radiation was found to consist of
two distinct types, one of which was cut off by five or six millimeters
of lead while the other possessed such great penetrating power
that its presence could be readily detected after passing through a
layer of lead fifteen centimeters thick.

Rutherford named these three distinct types of radiation, the
a-, 0-, and 7-rays, respectively. The penetrating powers of the
«-, 0-, and 7-rays may be approximately expressed by the propor-
tion 1 to 100 to 10,000; i.e., the 0-rays are 100 times more pene-

RADIOACTIVITY 47

trating than the a-rays, while the 7-rays are 100 times more
penetrating than the 0-rays.

The general characteristics of the three kinds of rays may be
briefly summarized as follows: —

(1) a-Rays. The a-rays consist of positively charged particles
moving with speeds approximately one-tenth as great as that
of light. These particles have been shown to be identical with
helium atoms carrying two positive charges of electricity. They
are appreciably deflected from a rectilinear path by magnetic and
electric fields. They possess great ionizing power but relatively
little penetrating power or photographic action. The depth to
which an oc-particle penetrates a homogeneous absorbing medium
before losing its ionizing power, is known as its " range." The
range is proportional to the cube of the initial speed of the a-
particle and is one of the characteristic properties of the radio-
elements emitting a-rays.

(2) 0-Rays. The 0-rays consist of negatively charged particles
moving with speeds varying from two-fifths to nine-tenths of the
speed of light. They are, in fact, electrons moving with much
greater speeds than those shot out from the cathode in a vacuum
tube. While the a-particles emitted by a particular radio-element
have a definite velocity, the corresponding 0-ray emission consists
of a flight of particles having widely different speeds. The pene-
trating power of the 0-rays is conditioned by the speed of the
particles, those which move most rapidly possessing the greatest
penetrating power.

The ionizing action of the 0-rays is much weaker than that of the
a-rays, while exactly the reverse is true of the photographic action.

(3) y-Rays. The 7-rays are identical with X-rays. They con-
sist of extremely short waves of light, the wave-length varying
from about 1 X 10~8 cm. for the rays of low penetrating power to
about 1 X 10~9 cm. for the most penetrating rays. Obviously the
7-rays cannot be deflected from a rectilinear path by either electric
or magnetic fields.

Uranium-X and Thorium-X. In 1900, Crookes * precipitated
a solution of a uranium salt with ammonium carbonate; when an

* Proc. Roy. Soc., 64, 409 (1900).

48 THEORETICAL CHEMISTRY

excess of the reagent had been added, all but a minute portion of
the precipitate was found to have dissolved. This small insoluble
residue, though chemically free from uranium, was found, when
tested photographically, to be several hundred times more active,
weight for weight, than the original salt. The solution, on the
other hand, was found to have lost nearly all of its activity. At
the end of a year, however, the solution had entirely regained its
original activity, while the insoluble residue had become inactive.
The active substance thus separated was called, on account of its
unknown nature, uranium-X.

Similarly, Rutherford and Soddy,* by precipitating a solution
of a thorium salt with ammonium hydroxide, found that a large
proportion of the activity remained behind in the thorium-free
filtrate. On evaporating the filtrate to dryness and driving off
the ammonium salts, a residue was obtained which was, weight
for weight, several thousand times more active than the original
solution. After standing for a month, this residue was found to
have lost its Activity, while the precipitate had regained the ac-
tivity of the original thorium compound. This active residue was
called thorium-X from analogy to Crookes' uranium-X.

The fact that uranium-X and thorium-X had each been obtained
as the result of specific chemical processes, seemed to warrant the
conclusion that they are new substances possessing well-defined
properties. The manner in which these substances were obtained
led to a variety of speculations as to the mechanism of the process
involved in their production. In a subsequent paragraph it will
be shown that the so-called disintegration theory offers a most
satisfactory explanation of the foregoing experimental results.

The Emanations. The element thorium was found by Ruther-
ford to give off a radioactive gas or emanation which, when left tc
itself, rapidly loses its activity in a similar manner to uranium-X
and thorium-X. The thorium emanation was found to resemble
the inactive gases in its chemical behavior. Thus, it can be sub-
jected to the action of lead chromate, metallic magnesium and
zinc dust at extremely high temperatures without undergoing
change* The only other substances which, at the time of the dis-
* Phil. Mag., VI, 4, 370 (1902).

RADIOACTIVITY 49

covery of the emanation, were known to resist the action of these
reagents under the same conditions were the gaseous elements
helium, neon, argon, krypton, and xenon. The most conclusive
evidence of the gaseous character of the thorium emanation is the
fact that it condenses to a liquid at very low temperatures.

Rutherford and Soddy showed that the origin of the thorium
emanation is thorium-X and not the element thorium itself.
Freshly precipitated thorium hydroxide shows only a trace of
emanating power, whereas the thorium-X separated from the
filtrate possesses this power to a marked degree. As the thorium-
X gradually loses its emanating power, the hydroxide shows a
corresponding recovery.

Radium was found to give off an emanation which behaves
similarly to the thorium emanation except that it parts with its
activity at a slower rate. The rate at which the radium emanation
is produced, together with its longer life, enabled Ramsay and
Soddy * not only to measure the volume of the emanation obtained
from 60 mg. of radium bromide but also to establish the fact that
it obeys Boyle's law.

The Active Deposits. It was discovered^ by Rutherfordf for
thorium, and by M. and Mine. Curie t for radium, that the emana-
tions from these elements are ca-gable of imparting radioactivity
to surrounding objects. On the other hand, uranium and polo-
nium, which evolve no emanations, have no such influence on their
environment. This fact is taken as a proof that induced radio-
activity is due to actual contact with the emanations. If a nega-
tively charged platinum wire be exposed to the thorium or radium
emanation, the whole of its activity will be concentrated on the
wire. This so-called active deposit may be transferred from the
wii-e to a piece of sandpaper by rubbing. It may be sublimed
from the wire to the walls of the tube in which it is heated, or it-
may be dissolved from the wire by means of hydrochloric or sul-
phuric acid. On evaporating the solution, the activity will be found
to reside on the evaporating dish. Microscopic examination of

* Proc. Roy. Soc., 73, 346 (1904).
t Phil. Mag., VI, 23, 621 (1911).
i Coirmt. rendus. 120. 714 (1899).

50 THEORETICAL CHEMISTRY

the wire fails to reveal any deposit, and the most sensitive balance
is incapable of detecting any gain in weight after exposing the wire
to the emanation. These experiments leave no room for doubt
that the active deposits consist of infinitesimal amounts of solid
substances possessing definite chemical properties. The active
deposits undergo a gradual loss of activity similar to that observed
with the emanations and with uranium-X arid thorium-X.

The Disintegration Hypothesis. It was soon discovered that
the active deposits undergo a series of additional radioactive
changes. These subsequent transformations were found to be
much more obscure and difficult to follow. In fact, it is only be-
caiiife'of the ingenuity and mathematical acumen of those who
undertooK^Jbhis difficult research that we are today in possession of
such complete knowledge of the succeeding members of the radio-
active serie& of elements.

In*HK)3, Rutherford and Soddy * brought forward their theory
of atoi^^fcijtegration which affords a perfectly satisfactory in-
t&pjretatjg njp the complicated results already detailed, as well as
ofthe-ewbafequent changes in the active deposits to which reference
has just been made. According to this theory, radioactive change
is assumed to be due to the spontaneous disintegration of the radio-
elements with concomitant formation of new elements. These
new elements, which are often less stable than the parent element,
are assumed to undergo further disintegration with the production
of still other elements, the end of the process ultimately being
reached when a stable element is formed.

The Radioactive Constant. The activity of a radio-element
decays exponentially with time according to the equation —

It = Ioe~", (1)

where Jo is the initial activity, It the activity at the end of an in-
terval of time t, X a constant, and e is the base of the natural system
of logarithms. The constant X, known as the radioactive constant,
represents the fraction of the total amount of radioactive substance
undergoing disintegration in unit of time, provided the latter is so
small that the quantity at the end of time unit is only slightly
* Phil. Mag., VI, 5, 576 (1903).

RADIOACTIVITY 51

different from the initial quantity. The reciprocal of the radio-
active constant is called the average life of the element. Soddy
defines the average life of a radio-element as "the sum of the sepa-
rate periods of future existence of all the individual atoms divided
by the number in existence at the starting point."

If nt represents the number of atoms of a radio-element changing
in unit time at the end of a time t, and no the corresponding value
when t = 0, equation (1) may be written

Tit == no6~~ •

In order to determine the initial rate of change, let N0 denote
total number of atoms originally present, and Nt the nu
maining unchanged at time t] we then have

*-/

But when t = 0, NQ = Nt.

and No = *

Hence Nt

On differentiating, we have

f--xy. (2)

Or, stated in words, the rate at which the atoms of a radio-element
undergo disintegration at any given time is found to be propor-
tional to the total number in existence at that time.

This law of radioactive change is also peculiar to unimolecular
chemical reactions (see p. 361). The velocity of a unimolecular
reaction, however, is conditioned by the temperature, whereas
the velocity of a radioactive change remains unaltered at the
highest and the lowest attainable temperatures.

The time required for one-half of a radio-element to undergo
transformation is known as the period of half change T, and may be
readily calculated from X as follows: —
log 0.5 = 0.4343 \T

m 0.6932

«'i — - — •
* — \

52 THEORETICAL CHEMISTRY

It has been shown by Geiger and Nuttall * that the radioactive
constant X and the range R of the a-particles shot out from a dis-
integrating atom bear the following empirical relation to each
other,

X = aR\

where a and b are constants, the former constant being character-
istic of the particular radioactive series to which the element be-
longs. This formula has been found useful in calculating the
values of X for the longest and shortest lived elements.

Radioactive Equilibrium. It is evident that a state of equi-
librium must ultimately be attained among the atoms of a radio-
active substance. When the rate of production of a radio-element
from its parent element is equal to its rate of disintegration into
the next succeeding element of the series, the substance is said to
be in radioactive equilibrium. The relative amounts of the suc-
cessive members of a series of elements in radioactive equilibrium
are inversely proportional to their radioactive constants.

In order that measurements of the rate of radioactive change
may be strictly comparable, it is necessary to make use of the
amounts which are in equilibrium with a fixed amount of the
parent element. Thus, the unit adopted for the measurement of
the quantity of the radium emanation is the mass of emanation in
equilibrium with one gram of radium. This unit is known as the
curie. Its mass is Xi/X2 gram, where Xi and X2 are the radioactive
constants of radium and its emanation respectively. One curie
of radium emanation may be shown to occupy 0.63 cu. mm. under
standard conditions of temperature and pressure.

The Disintegration Series. It appears almost certain that the
thirty or more radio-elements are disintegration products of one or
the other of the two parent elements, uranium or thorium. One
of the most convenient methods of classification of these elements
is to arrange them in disintegration series, starting with the parent
element and placing the succeeding elements in the order of their
production. The accompanying table shows the three series of
radio-elements as thus arranged by Soddy. The numbers within
the circles are the atomic weights of the elements, while the small
* Phil. Mag., VI, 22, 613 (1911).

RADIOACTIVITY
TABLE SHOWING DISINTEGRATION SERIES

54 THEORETICAL CHEMISTRY

circles and dots at the right of the larger circles indicate the char-
acter of the radiation given out at each stage of the disintegration
process. The average life, 1/X, of each element in the series is
given below the name of the element. While a detailed account
of the properties of the different radio-elements included in this
table cannot be given here, attention should be called to the com-
plex transformations occurring in the active deposit. It is also of
interest to observe that the end-products of the three series bear a
striking resemblance to the element lead and that their atomic
weights are approximately equal and nearly identical with that
of ordinary lead.

Counting the a-Particles. It has already been stated that
when an a-particle strikes a screen coated with phosphorescent
zinc sulphide, a distinct flash of light may be seen when the screen
is viewed through a magnifying lens in a dark room. It is obvious
that if one could count the number of these scintillations, it would
be an easy matter to ascertain the total number of a-particles
shot out from a radioactive substance in a given time. By using
a phosphorescent screen and a microscope, Regener * has deter-
mined in this manner the rate of emission of a-particles from
polonium. He found from his different experiments an average
emission of 3.94 X 105 a-particles per second. The total charge
on the a-particles was then measured by collecting them in a suit-
able measuring vessel. A charge of 37.7 X 10~~5 electrostatic
units was found to be associated with the total number of a-
particles emitted by polonium in one second.

Rutherford and Geiger f devised an electrical method for count-
ing the a-particles. In their experiment the source of the a-
particles was a small disc which had been exposed to the radium
emanation for some hours. This disc was placed in an evacuated
tube at a measured distance from a small aperture of known cross-
section. The aperture was closed with a thin plate of mica through
which the a-particles could pass with Qase. After passing through
th$ mica plate, the a-particles entered an ionization chamber
filled with air at reduced pressure and fitted with two charged

* Sitzunsbericht d. K. preuss. Akad., 38, 948 (1909).
t Proc.Boy. Soc. A, 8x, 141 (1908),

RADIOACTIVITY 55

metal plates connected with appropriate apparatus for measuring
ionization currents. Whenever an a-particle entered the ioniza-
tion chamber, a momentary current passed, producing a sudden
deflection of the needle of the electrometer. By counting the
number of throws of the needle occurring in a definite interval of
time, the total number of a-particles passing through the ionization
chamber was determined. Knowing the distance of the source
of the radiations from the aperture, together with the area of the
aperture, the total number of a-particles emitted by the radio-
active disc in a given time could be computed. Rutherford and
Geiger thus found that one gram of radium emits very nearly
107 X 1016 a-particles per year. Having determined the total
number of a-particles emitted, it only remained to measure the
total charge, in order to calculate the charge carried by a single
a-particle. From a series of very consistent measurements, the
charge carried by a single a-particle was found to be 9.3 X 10~~10
electrostatic units. Since the fundamental charge e has been
shown to be 4.48 X 10~10 electrostatic units, it follows that the
a-particle carries two ionic charges of electricity.

Helium Atoms and a-Particles Identical. In 1909, Ruther-
ford and Royds * performed a crucial experiment to determine the
nature of the a-particle. A glass bulb was blown with walls thin
enough to permit the passage of the a-particles but sufficiently
strong to withstand atmospheric pressure. The bulb was filled
with radium emanation and then enclosed in an outer glass tube
to which a spectrum tube had been sealed. » On exhausting the
outer tube and examining the spectrum of the residual gas, no
evidence of helium was obtained until after an interval of twenty-
four hours. After four days the characteristic yellow and green
lines were plainly visible and at the end of the sixth day, the com-
plete spectrum of helium was obtained. The unavoidable
elusion from this experiment is, that the presence of helium ir
outer tube must have been due to the a-particles whiplr were
projected through the thin walls of the inner tube. Jn anothfli
experiment, the inner tube was filled with pure/^helium uifoer
pressure while the exhausted outer tube was exaifuned fop^elium,

* Phil. Mag., VI, 17, 281

56 THEORETICAL CHEMISTRY

No trace of helium could be detected spectroscopically even after
an interval of several days, thus proving that the helium detected
in the first experiment must have resulted from the a-particles
which had been shot out from the radium emanation with sufficient
energy to penetrate the thin walls of the inner tube. These experi-
ments leave no room for doubt that an a-particle becomes a
helium atom when its positive charge is neutralized.

Rate of Production of Helium. The rate of production of
helium from the series in equilibrium with one gram of radium
has been determined experimentally by Rutherford and Bolt-
wood * to be 156 cu. mm. per year. This result agrees closaly
with the calculated rate of production, viz., 158 cu. mm. per gram
of radium per year.

Energy Evolved by Radium. Curie and Laborde * were the
first to call attention to the interesting fact that the temperature
of radium compounds was uniformly higher than that of their
environment. Careful measurements have shown that one, gram
of radium evolves heat at the rate of approximately 135 gram-
calories per hour. That the greater part of this heat energy is due
to the a-particles may be proven by a direct calculation of their
mean kinetic energy. The magnitude of the store of energy con-
tained in radium may be realized upon the statement/, that one
gram of radium; before it entirely disappears, evolves an amount
of heat energy nearly one million times greater than that evolved
in the formation of one gram of water from its elements.

* Phil. Mag., VI, 22, 586 (1911).
t Compt. rendus, 136, 673 (1904).

CHAPTER V.

ATOMIC STRUCTURE.

The Modern Conception of Atomic Structure. As a result of
the investigations of Thomson,* Rutherford,t Nicholson,} Bohr,§
and others, a theory of atomic structure has been developed which
affords a satisfactory interpretation of many of the important
relationships among the chemical elements.

Briefly stated, this theory assumes that the atom consists of a
central, positively charged nucleus, surrounded by a miniature
solar system of electrons. The investigations of Rutherford and
Geiger|| show that the character of the deflection of a-particles
shot out from radioactive atoms at speeds approximating 20,000
miles per second, and consequently completely penetrating other
atoms, is sucBTSg~to indicate an extremely high concentration of
positive electricity on the central nucleus. The central nucleus
which is supposed to represent nearly the entire mass of the atom,
is thought to be very small in comparison with the size of the atom
as a whole. Recent investigations make it appear probable that
the maximum diameter of the nucleus of the hydrogen atom is
about one one-hundred-thousandth of the diameter commonly
attributed to the atom. In commenting on this statement, Har-
kins says: 1f — "On this basis the atom would have a volume a
million-billion times larger than that of its nucleus, and thus the
nucleus of the atom is much smaller in comparison with the size
of the atom than is the sun when compared with the dimensions,
of its planetary system." It is highly probable that the central
nucleusjg itself made up of a definite number of units of positive
electricity together with a small number of attendant electrons.

* Phil. Mag., 7, 237 (1904).
t Popular Science Monthly, 87, 105 (1915)
t Phil. Mag., 22, 864 (1911).
§ Phil. Mag., 26, 476, 857 (1913).
|| Phil. Mag., 21, 669 (1911).
If Science, 66, 419 (1917).
57

58 THEORETICAL CHEMISTRY

It is further assumed that the units of positive electricity are
hydrogen atoms, each of which has been deprived of one electron.
If the mass of an atom is largely due to the presence of hydrogen
nuclei, then we should expect Prout's hypothesis to hold and the
atomic weights of the elements to be exact multiples of the atomic
weight of hydrogen. When we consider, however, that according
to electromagnetic theory the total mass of a body composed of
positive and negative units is dependent upon the relative posi-
tions of these units when packed together, it is evident that the
mass of the atom will not necessarily be an exact multiple of the
mass of the hydrogen atom.

It has already been pointed out that helium is a product of
many radioactive transformations. This fact may be taken as an
indication of the extraordinary stability of the helium atom. Be-
cause of its stability, the nucleus of this atom has come to be con-
sidered as a secondary unit of positive electricity. The nucleus
of the helium atom, or the nucleus of an a-particle is assumed to
consist of four hydrogen nuclei with two nuclear electrons.

The Atomic Number. Since the algebraic sum of the positive
and negative electrification on an atom must be zero, it follows
that the charge resident upon the nucleus must be equal to the
number of electrons outside the nucleus. This number, which has
come to be recognized as more important and characteristic than
the atomic weight, is known as the atomic number.

X-Rays and Atomic Structure. The discovery by W. L. Bragg
in 1912 that X-rays undergo reflection at crystal surfaces and the
subsequent development by Mr. Bragg and his father, W. H. Bragg
of the X-ray spectrometer, has led to a series of investigations of
the utmost importance to both the chemist and the physicist.

In order that the significance of these investigations may be
understood it may be well to summarize very briefly some of the
more important properties of the X-ray.

The bombardment of metal plates, usually of platinum, by elec-
trons give rises to X-rays. The radiation issuing from an X-ray
tube is very far from homogeneous. When screens of different
materials and varying thicknesses are interposed in the path of
the rays, the degree of absorption is irregular. It has been found,

ATOMIC STRUCTURE 59

however, that every substance when properly stimulated is capable
of emitting a homogeneous and characteristic X-radiation, the
penetrating power of which is wholly determined by the nature of
the elements of which the substance is composed. The pene-
trating power of this typical X-radiation increases with the atomic
weight of the radiating element. With elements whose atomic
weights are less than 24, the radiation is too feeble to be measured.
It is important to note that this property of the elements is not a
periodic function of the atomic weight. This type of X-radiation
is entirely independent of external conditions, indicating that it
is closely connected with the internal structure of the atoms from
which it emanates. The rays possess the power of affecting the
photographic plate and also of rendering gases through which
they pass conductors of electricity.

It is estimated that the wave-length of an X-ray is about 1 X 10~8
to 1 X 10~9 cm., or about one ten-thousandth of the wave-length
of sodium light. It is obvious that the spacing of the lines of a
grating capable of diffracting such short waves must be of the
order of magnitude of interrnolecular distances. It is well known
that a grating owes its power of analyzing a complex system of
light waves into its component wave-trains, to the series of paral-
lel lines which are ruled upon its surface at exactly equal intervals.
When a train of waves is incident upon a grating, each line acts
as a center from which a diffracted train of waves emerges.

Such a grating is relatively simple in its action since it consists
of a single series of centers of diffraction lying in one plane. The
power of a crystal surface to reflect X-rays, however, is due to the
fact that the crystal is in reality a three-dimensional diffraction
grating, the atoms or molecules of which the crystal is built up,
acting as the centers of diffraction. It must be borne in mind that
the reflection of X-rays is in no way dependent upon the existence
of a polished surface on the outside of the crystal, but rather upon
the regularly spaced atoms or molecules within the crystal. To
ordinary waves of light the atomic structure is so fine grained as
to behave as a continuous medium, whereas to the short X-ray
waves, the crystal acts as a discontinuous structure of regularly
arranged particles, each of which functions as a diffraction center.

THEORETICAL CHEMISTRY

X-Ray Spectra. By making use of the reflecting power of one
of the cleavage planes of a crystal, and employing different metals
as anti-cathodes in an X-ray tube, Moseley * succeeded in photo-
graphing the X-ray spectra of the characteristic radiations of a
number of the elements. He showed that the X-ray spectrum of
an element is extremely simple and consists of two groups of lines
known as the "K" and "L" radiations. As a result of careful
study of the "K" radiations of thirty-nine elements from alumin-
ium to gold, Moseley discovered that these radiations are char-
acterized by two well-defined lines whose vibration frequency v
is connected with the atomic number of the clement N, by the
simple relation v = A (N — IY

where A is a constant. When the square roots of the frequencies
of the elements are plotted as abscissae against their atomic num-
bers as ordinates,
the points are found
to lie on a straight
line as shown in Fig.
11. On the other
hand, if the square
roots of the fi&quen-
cies are plotted
against the atomic
weights of the ele-
ments, the relation-
ship is no longer rec-
tilinear. When the
elements are ar-
ranged in the order
of their atomic num-
bers instead of in the
order of their atomic weights, the irregularities f hitherto noted in
connection with argon, cobalt, and tellurium entirely disappear.
In reviewing Moseley's work on X-ray spectra, Soddy t says: —

* Phil. Mag., 26, 210 (1913); 27, 703 (1914).

t See p. 29.

j Ann. Reports on the Prog, of Chemistry, p. 278 (1914).

fin

100

•f

80 Jc

*x^

60 5

+ V^

-i;

+*s

«0

10 U 18

Square Root of Frequency X

Kg. 11.

ATOMIC STRUCTURE 61

"A veritable roll-call of the elements has been made by this
method. Thirty-nine elements, with atomic weights between
those of aluminium and gold, have been examined in this way, and
in every case the lines of the X-ray spectrum have been found to
be simply connected with the integer that represents the place
assigned to it by chemists in the periodic table."

One of the most interesting results of this " roll-call" of the
elements is the fixing of the number of possible rare-earth elements.
Between barium and tantalum there are places for only fifteen
rare-earth elements and fourteen of these places are filled. While
future investigation may necessitate some rearrangement in the
order of tabulation, the total number of these elements is limited
to fifteen.

Periodicity among the Radio-elements. The problem of plac-
ing the newly discovered radio-elements in the periodic table re-
mained unsolved until 1913, when Fajans * and Soddy,t working
independently, discovered an important generalization concerning
the changes in chemical properties resulting from the expulsion of
a- and jft-particles during radioactive transformations. This impor-
tant generalization may be stated as follows: — The expulsion of
an a-$article causes a radioactive element to shift its position in
the periodic table two places in the direction of decreasing atomic
weight, whereas the emission of a fi-particle causes a shift of one
place in the opposite direction. This generalization not only agrees
with our present theory of atomic structure, but may be shown
to be a necessary consequence of this theory.

The loss of an a-particle or helium atom involves a loss of 4
units in atomic weight and of 2 units of positive electricity from
the nucleus of the atom. In consequence of this loss, the atomic
number is diminished by 2 units and the resulting new element
will find a place in the periodic table two groups to the left of that
occupied by the parent element. On the contrary, while the
expulsion of a 0-particle, or electron, involves practically no change
in mass, the nucleus of the parent atom suffers a loss of 1 unit of
negative electricity. This loss is equivalent to a gain of 1 unit of

* Physikal. Zeit., 14, 49 (1913).
t Chem. News, 1-07, 97 (1913).

THEORETICAL CHEMISTRY

positive electricity, or to an increase of 1 unit in the atomic number,
and in consequence, the position of the new element in the periodic
table will be shifted one group to the right of that occupied by the
parent element.

Soddy's arrangement of all of the radio-elements in accordance
with this generalization is shown in Fig. 12. Thus, starting with
the element uranium in Group VIA, we may follow the successive

RADIO-ELEMENTS AND PERIODIC LAW
ALL ELEMENTS IN THE SAME PLACE

IN THE PERIODIC TABLE
ARE CHEMICALLY NON-SEPARABLE
^V AND (PROBABLY)

>§PECTROSCOPICALLY INDISTINGUISHABLE

Relative No. of Negative Electrons
5 4 82 1

Fig. 12.

steps in the radium disintegration series which was discussed in
the preceding chapter. The element UXi resulting from U by
the loss of an a-particle is placed in Group IVA. This element in
turn undergoes a 0-ray change producing the element UX2 which
is accordingly placed in Group VA. The element UII in Group
VIA is formed from UX2 by #-ray disintegration, while the ele-
ment lo results from the loss of an a-particle by UII with a con-
sequent shifting of two places to the left in the table. A similar
loss of an a-particle by lo brings us to the element Ra in Group

ATOMIC STRUCTURE 63

IIA. In the successive steps of this disintegration from U to Ra,
three a-particles or 12 units of atomic mass are lost, and the atomic
weight of Ra, as calculated from that of U, agrees with the atomic
weight found by direct experiment. In like manner the remain-
ing stages of the disintegration may be followed to the end-product
in Group IVB.

Isotopes. Perhaps the most striking feature in the table is
the occurrence of several different elements in the same place, as
for example in Group IVB, where in the place occupied by the
element Pb, we also find RaB, RaD, ThB, and AcB, together
with four other elements to which no names have been assigned,
but which are none the less stable end-products. The individual
members of such a group of elements occupying the same place in
the periodic table, and being in consequence chemically identical,
are known as isotopes. Isotopic elements have identical arc and
spark spectra and, except for differences in atomic weight, are
chemically indistinguishable.

Making use of the fact that two gaseous elements having differ-
ent atomic weights diffuse at different rates, Thomson and Aston
have recently succeeded in separating neon into two isotopes
having atomic weights 20 and 22 respectively. This is the only
method which has thus far given promise of success in effecting
isotopic separations.

It is interesting to note in Soddy's table (Fig. 12), that "the ten
occupied spaces (groups) contain nearly forty distinct elements,
whereas if chemical analysis alone had been available for their
recognition, only ten elements could have been distinguished."

The Hydrogen-Helium System of Atomic Structure. A
generalization similar to that just outlined for the radio-elements
has been found by Harkins and Wilson * to hold true for the lighter
elements which apparently do not undergo appreciable a-ray dis-
integration. Beginning with helium and adding 4 units of atomic
weight for each increase of 2 units in the atomic number, gives th^
atomic weights of the elements in the even-numbered groups of
the periodic table, neglecting small changes in mass due to nuclear
packing. This rule has been found to hold very closely for all of
the elements having atomic weights below 60.

* Proc. Nat. Acad., Vol. I, p. 276 (1915),

64 THEORETICAL CHEMISTRY

The atomic weights of the elements of the odd-numbered groups
can be calculated by a similar rule, provided that the atom of
lithium, the first member of the odd-numbered groups, be assumed
to be made up of 1 hydrogen and 3 helium nuclei. The following
table gives the results as calculated by Harkins and Wilson for the
first three series of the periodic table.

The so-called theoretical atomic weights are calculated on the
basis H = 1, while the experimentally determined values are on
the basis O = 16 or H = 1.0078. The remarkably close agree-
ment between the two sets of values is taken as an indication that
the packing effect, resulting from the formation of the elements
from hydrogen nuclei and attendant electrons, is very small.
This packing effect has been estimated to involve a decrease in
atomic mass of about 0.77 per cent, and is believed to be due
almost entirely to the formation of the helium atom. The hydro-
gen-helium hypothesis of atomic structure offers a rational ex-
planation of many interesting but hitherto obscure facts concern-
ing the nature of the elements.

Relation between Atomic Weights and Atomic Numbers. For
all elements whose atomic weights are less than that of nickel,
Harkins finds the following simple mathematical relation to hold,

where W is the atomic weight and N is the atomic number. In
other words, the atomic weights are a linear function of the atomic
numbers.

The Periodic Law. In the light of recent discoveries the
periodic law acquires new significance; in fact to-day the periodic
law may be regarded as the most comprehensive generalization in
the whole science of chemistry.

Attention has already been directed in an earlier chapter to the
most apparent of the imperfections in Mendeteeff s system of
classification of the elements. While the later tables are more
complete than the original, owing in part to the discovery of new
elements, it must be admitted nevertheless that relatively little
real progress has been made until recently toward removing the
seemingly inherent defects of the system.

ATOMIC STRUCTURE

ooO*88

MM
00

CSI

03 O) O O

°-

,08+00

ffi S QO

^00

W
+00

•4J

66 THEORETICAL CHEMISTRY

A satisfactory periodic table should meet the following require-
ments: —

(1) It should afford a place for isotopic elements such as lead.

(2) The radio-elements together with their a- and j3-disinte-
gration products should be shown.

(3) It should contain no vacant spaces except those correspond-
ing to the atomic numbers of undiscovered elements.

(4) It should bring out the relation between the elements con-
stituting a main group and those forming the corresponding sub-
group. For example, the relation between the elements Be, Mg,
Ca, Sr, Ba, and Ha on the one hand, and the elements Zn, Cd, and
Hg on the other, should be emphasized.

(5) The elements of Group O and Group VIII should fit natu-
rally in the table.

(6) All of the foregoing conditions should be shown by means of
a continuous curve connecting the elements in the order of their
atomic numbers, the latter having been shown to be more charac-
teristic of an element than its atomic weight.

A table which satisfactorily meets these requirements has re-
cently been devised by Harkins and Hall. This table may be
constructed in the form of a helix in space or as a spiral in a plane.
The following description of the helical arrangement, shown in
Fig. 13, is taken verbatim from the original paper of Harkins and
Hall.*

"The atomic weights are plotted from top down, one unit of
atomic weight being represented by one centimeter, so the model
is about two and one-half meters high. . . .

' 4 The balls representing the elements are supposed to be strung
on vertical rods. All of the elements on one vertical rod belong to
one group, have on the whole the same maximum valence, and are
represented by the same color. The group numbers are given at
the bottom of the rods. On the outer cylinder the electro-nega-
tive elements are represented by black circles at the back of the
cylinder, and electro-positive elements by white circles on the front
of the cylinder. The transition elements of the zero and fourth
groups are represented by circles which are half black and half

* Jour. Am. Chem. Soc., 38, 169 (1916).

100

120

140

160

180

200

220

240

Group

Fig. 13.

68 THEORETICAL CHEMISTRY

white. The inner loop elements are intermediate in their proper-
ties. Elements on the back of the inner loop are shown as heavily
shaded circles, while those on the front are shaded only slightly.
"In order to understand the table it may be well to take an
imaginary journey down the helix, beginning at the top. Hydro-
gen (atomic number and atomic weight = 1) stands by itself, and
is followed by the first inert, zero group, and zero valent element
helium. Here there comes the extremely sharp break in chemical
properties with the change to the strongly positive, univalent
element lithium, followed by the somewhat less positive bivalent
element, beryllium, and the third group element boron, with a
positive valence of three, and a weaker negative valence. At the
extreme right of the outer cylinder is carbon, the fourth group
transition element, with a positive valence of four, and an equal
negative valence, both of approximately equal strength. The
first element on the back of the cylinder is more negative than
positive, and has a positive valence of five, and a negative valence
of three. The negative properties increase until fluorine is reached
'and then there is a sharp break of properties, with the change from
the strongly negative, univalent element fluorine, through the zero
valent transition element neon, to the strongly positive sodium.
Thus in order around the outer loop the second series of elements
are as follows: —

Group number 01234567

Maximum valence. .. 01234567

Element He Li Be B C N 0 F

Atomic number 2 3 4 5 6 7 89

" After these comes neon, which is like helium, sodium which is
like lithium, etc., to chlorine, the eighth element of the second
period. For the third period the journey is continued, still on
the outer loop, with argon, potassium, calcium, scandium, and-
then begins with titanium, to turn for the first time into the inner
loop. Vanadium, chromium, and manganese, which comes next,
are on the inner loop, and thus belong, not to main but to sub-
groups. This is the first appearance in the system of sub-group
elements. Just beyond manganese a catastrophe of some sort
seems to take place, for here three elements of one kind, and there-

ATOMIC STRUCTURE 69

fore belonging to one group, are deposited. The eighth group in
this table takes the place on the inner loop which the rare gases of
the atmosphere fill on the outer loop. The eighth group is thus a
sub-group of the zero group.

" After the eighth group elements, which have appeared for the
first time, come copper, zinc, and gallium; and with germanium, a
fourth group element, the helix returns to the outer loop. It then
passes through arsenic, selenium, and bromine, thus completing
the first long period of 18 elements. Following this there comes a
second long period, exactly similar, and also containing 18 elements.

"The relations which exist may be shown by the following
natural classification of the elements. They may be divided into
cycles and periods as follows:

TABLE I.

Cycle 1 = 42 elements.

1st short period He — F = 8 = 2X22 elements.

2nd short period Ne — Cl = 8 = 2X22 elements.

Cycle 2 = 62 elements.

1st long period A — Br = 18 = 2 X 32 elements.

2nd long period Kr — I «= 18 = 2 X 32 elements.

Cycle 3 = 82 elements.

1st very long period Xe — Eka-I = 32 = 2 X 42 elements.

2nd very long period Nt — U.

" The 'last very long period, and therefore the last cycle, is in-
complete. It will be seen, however, that these remarkable relations
are perfect in their regularity. These are the relations, too, which
exist in the completed system,* and are not like many false nu-
merical systems which have been proposed in the past where the
supposed relations were due to the counting of blanks which do not
correspond to atomic numbers. This peculiar relationship is un-
doubtedly connected with the variations in structure of these com-
plex elements, but their meaning will not be apparent until we
know more in regard to atomic structure.

* If elements of atomic weights two and three are ever discovered then the
zero cycle would contain 2P elements, and period number one should then be
said to begin with lithium. Such extrapolation, however, is an uncertain
basis for the .prediction of such elements.

70 THEORETICAL CHEMISTRY

"The first cycle of two short periods is made up wholly of outer
loop or main group elements. Each of the long periods of the
second cycle is made up of main and of sub-group elements, and
each period contains one-eighth group. The only complete very
long period is made up of main and of sub-group elements, con-
tains one-eighth group, and would be of the same length (18 ele-
ments) as the long periods if it were not lengthened to 32 elements
by the inclusion of the rare earths.

"The first long period is introduced into the system by the in-
sertion of iron, cobalt, and nickel, in its center, and these are three
elements whose atomic numbers increase by steps of one while
their valence remains constant. The first very long period is
formed in a similar way by the insertion of the rare earths, another
set of elements whose atomic numbers increase by one while the
valence remains constant.

"In this periodic table the maximum valence for a group of
elements may be found by beginning with zero for the zero group
and counting toward the front for positive valence, and toward
the back for negative valence.

"The negative valence runs along the spirals toward the back
as follows: —

0 -1 -2 -3 -4

Ne F O N C

A Cl S P Si

"Beginning with helium the relations of the maximum theoreti-
cal valences run as follows: —

Case 1 . He-F 0, 1, 2, 3, 4, 5, 6, 7, but does not rise to 8. Drops by 7 to 0.

Ne-Cl ... 0, 1, 2, 3, 4, 5, 6, 7, but does not rise to 8. Drops by 7 to 0.
Case 2. A-Mn ... 0, 1, 2, 3, 4, 5, 6, 7, 8, 8. Drops by 7 to 1.

Fe, Co, Ni.
Case 1. Cu-Br.
Case 2. Kr-Ru, Rh, Pd.

"In the third increase, the group number and maximum valence
of the group rise to 8, three elements are formed, and the drop is
again by 7 to 1.

"Thus in every case when the valence drops back the drop in
maximum group valence is 7, either from 7 to 0, or from 8 to 1.

ATOMIC STRUCTURE 71

This is another illustration of the fact that the eighth group is a
sub-group of the zero group. The valence of the zero group is
zero. According to Abegg the contra-valence, seemingly not
active in this case, is eight.

"In Fig. 13 the table is divided into five divisions by four
straight lines across the base. These divisions contain the fol-
lowing groups: —

Division 0 1 2 3 4

Groups 0,8 1,7 2,6 3,5 4,4

"The two groups of any division are said to be complementary.
It will be seen that the sum of the group numbers in any division
is equal to 8, as is also the sum of the maximum valences. The
algebraic sum of the characteristic valences of two complementary
groups is always zero. In any division in which the group numbers
are very different, the chemical properties of the elements of the com-
plementary main groups are very different, but when the group
numbers become the same, the chemical properties become very much
alike. Thus the greatest difference in group numbers occurs in
division 8, where the difference is 8, and in the two groups there is
an extreme difference in chemical properties, as there is also in
division 1 between Groups 1 and 7»

f( Whenever the two main groups of a division are very different in
properties, each of the sub-groups is quite different from its related
main group. Thus copper in Group IB is not very closely related
to potassium Group IA in its properties, and manganese is not
very similar to chlorine, but as the group numbers approach each
other the main and sub-groups become much alike. Thus scandium
is quite similar to gallium in its properties, and titanium and ger-
manium are very closely allied to silicon.

"One important relation is that on the outer cylinder the main
groups I A, HA, III A, become less positive as the group number
increases, while on the inner loop the positive character increases from
Group IB to IIB, and at the bottom of the table the increase from
IIB to IIIB is considerable. Thus thallium is much more posi-
tive than mercury. It has already been noted that in the case of
the rare earths also the usual rule is inverted, that is the basic
properties decrease as the atomic weight increases.'1

CHAPTER VI.
GASES.

The Gas Laws. Matter in the gaseous state possesses the
property of filling completely and to a uniform density any avail-
able space. Among the most pronounced characteristics of
gases are lack of definite shape or volume, low density and small
viscosity. The laws expressing the behavior of gases under differ-
ent conditions are relatively simple and to a large extent are
independent of the nature of the gas. The temperature and
pressure coefficients of aliases are very nearly the same.
— KTT662, Robert Boyle discovered the familiar law that at
constant temperature, the volume of a gas is inversely proportional
to the pressure upon it. This may be expressed mathematically
as follows: —

v oc - (temperature constant)

where v is the volume and p the pressure.

In 1801, Gay-Lussac discovered the law of the variation of the
volume of a gas with temperature.

This law may be formulated thus: — At constant pressure, the
volume of a gas is directly proportional to its absolute temperature,
or

v <x T (pressure constant).

There are three conditions which may be varied, viz., volume,
temperature and pressure. The preceding laws have dealt with
the relation between two pairs of the variables when the third
is held constant. There remains to consider the relation between
the third pair of variables, pressure and temperature, the volume
being kept constant. Evidently a necessary corollary of the first
two laws is that at constant volume, the pressure of a gas is directly
proportional to its absolute temperature, or

p oc T (volume constant),
72

GASES 73

These three laws may be combined into a single mathematical
expression as follows: —

v oc - (T const.) law of Boyle,

v oo T (p const.) law of Gay-Lussac;
combining these two variations we have,

00C-,

P
or introducing a proportionality factor ik,

i T

V = # — 9
P

vp = kT. (1)

If the temperature of the gas be 0° (273° absolute), and the corre-
sponding volume and pressure VQ and p<> respectively, then (1)
becomes

z>o??o = 273 fc,
and

~ 273'
eliminating the constant k between (1) and (2), we have

- VQP°
-

For any one gas the term ~ is a constant. If VQ is the volume
of 1 gram of gas at 0° and 76 cm., we write

vp = rT, (3)

where v is the volume of 1 gram of gas at the temperature T and
the pressure p, and r is a constant called the specific gas constant.
On the other hand when v0 denotes the volume of one mol. of gas
at 0° and 76 cm. (22.4 liters), the equation becomes

vp = RT, (4)

where R is termed the molecular gas constant, which has the game

74 THEORETICAL CHEMISTRY

value for all gases. If M is the molecular weight of the gas, Mr
= R. Equation (4) is the fundamental gas equation.

Evaluation of the Molecular Gas Constant. Since the product
of p and v represents work, and T is a pure number, R must be
expressed in energy units. There are four different units in which
the molecular gas constant is commonly expressed, viz., (1) gram-
centimeters, (2) ergs, (3) calories, and (4) liter-atmospheres.

1. R in gram-centimeters. The volume, v, of 1 mol. of gas at
0° and 76 cm. is 22.4 liters or 22,400 cc. The pressure, p, is 76 cm.
multiplied by 13.59, (the density of mercury), or 1033.3 grams per
square centimeter. Substituting we obtain

poVQ 1033.3 X 22,400
# = -yT — - 273 - = 84>760 ST- cm-

2. R in ergs. To convert gram-centimeters into ergs we must
multiply by the acceleration due to gravity, g = 980.6 cm. per
sec. per sec., or

R = 84,760 X 980.6 = 83, 150,000 ergs.

3. R in calories. To express work in terms of heat, we must
divide by the mechanical equivalent of heat, or since, 1 calorie is
equivalent to 42,640 gr. cm. or 41,830,000 ergs, we have

83 150 000
R = / = *'®® ca*' (approximately 2 cal.)

4. R in liter-atmospheres. A liter-atmosphere may be defined
as the work done by 1 atmosphere on a square decimeter through
a decimeter. If p0 is the pressure in atmospheres, and VQ is the
volume in liters, we have

R = -JT == -0 = 0.0821 liter-atmosphere.

Deviations from the Gas Laws. Careful experiments by
Amagat * and others on the behavior of gases over extended ranges
of temperature and pressure have shown that the fundamental
gas equation, pv = RT, is not strictly applicable to any one gas,
the deviations depending upon the nature of the gas and the
conditions under which it is observed. It has been shown that
the gas laws are more nearly obeyed the lower the pressure, the
* Ann. Chim. phys. (5) 19, 345 (1880).

GASES

higher the temperature and the further the gas is removed from
the critical state. A gas which would conform to the require-
ments of the fundamental gas equation is called an ideal or per-
fect gas. Almost all gases are far from ideal in their behavior.
At constant temperature the product, pv, in the gas equation is
constant, so that if we plot pressures as abscissae and the corre-

100

P (atmospheres)
Fig. 14.

200

800

spending values of pv as ordinates, for an ideal gas we should
obtain a straight line parallel to the axis of abscissae, as shown in
Fig. 14. The results obtained by Amagat with three typical
gases are also shown in the same diagram. It will be apparent
that all of these gases depart widely from ideal behavior. In
the case of hydrogen pv increases continuously with the pressure,

76 THEORETICAL CHEMISTRY

while with nitrogen and carbon dioxide it first decreases, attains
to a minimum value and beyond that point increases with increas-
ing pressure. With the exception of hydrogen, all gases show a
minimum in the curve, thus indicating that at first the compress-
ibility is greater than corresponds with the law of Boyle, but
diminishes continuously until, for a short range of pressure, the law
is followed strictly: beyond this point the compressibility is less
than Boyle's law requires.

Hydrogen is exceptional in that it is always less compressible
than the law demands. This is true for all ordinary tempera-
tures, but it is highly probable that at extremely low temperatures
the curve would show a minimum. The two curves for carbon
dioxide at 31°.5 and 100° illustrate the fact that the deviations
from the gas laws become less as the temperature increases. The
deviations of gases from the laws of Boyle and Gay-Lussac, as well
as their behavior in general, may be satisfactorily accounted for
on the basis of the kinetic theory.

Kinetic Theory of Gases. The first attempt to explain the
properties of gases on a purely mechanical basis was made by
Bernoulli in 1738. Subsequently, through the labors of Kroenig,
Clausius, Maxwell, Boltzmann and others, his ideas were developed
into what is known today as the kinetic theory of gases. Accord-
ing to this theor^f gases are considered to be made up of minute,
perfectly elastic particles which are ceaselessly moving about
with high velocities, colliding with each other and with the walls
of the containing vessel. These particles are identical with the
molecules defined by Avogadro. The volume actually occupied
by the gas molecules is supposed to be much smaller than the
volume filled by them under ordinary conditions, thus allowing
the molecules to move about free from one another's influence
except when they collide. Tl^fi distance thrqugh which a molecule
moves before ftqll^ing with another molecule is known as its meg/£
freejQQtb. In terms of this theory, the pressure exerted by a gas
is due to the combined effect of the impacts of the moving molecules
upon the walls of the containing vessel, the magnitude of the
pressure beinff dependent upon the kinetic energy of the mole*
cules and their number^

GASES

Derivation of the Kinetic Equation. Starting with the assump-
tions already made, it is possible to derive a formula by means of
which the gas laws may be deduced. Imagine n molecules, each
having a mass, w, confined within the cubical vessel shown in
Fig. 15, the edge of which has a length, L While the different
molecules are doubtless Mpdving with different velocities, there
must be an average velocity for all of them. Let c denote this
mean velocity of translation. The molecules will impinge upon
tTRTwaJIiTin^ velocity of each may be resolved

according to the well-known dy namdcalp^ com-

Fig. 15.

ponents, a, y and z, parallel .to-the

and JL-JThe analytical expression for the velocity of a single
molecule, M, is

c2 = a* + y* + z\

f jUJ

In words, this means that the effect of the collision of the molecule
upon the wall of the containing vessel, is equivalent to the com-
bined effect of successive collisions of the molecule perpendicular
to the three walls of the cubical vessel with the velocities x, y and
z respectively! Fixing our attention upon the horizontal com-
ponent, the molecule will collide with the wall with a velocity x,
and owing to its perfect elasticity it will rebound with a velocity

78 THEORETICAL CHEMISTRY

— X, having suffered no loss in kinetic energy;. The momentum
before collision was mx and after collision it will be — mx, the
total change in momentum being 2 mx. The distance between
the two walls being I, the number of collisions on a wall in unit
time will be, x/l, and the total effect of a single molecule in one
direction in unit time will be 2 mx^x/l = 2 mx2/l. The same
reasoning is applicable to the other components, so that the com-
bined action of a single molecule on the six sides of the vessel
will be

2 m , 9 . 9 . 9. 2 me2

2 fYVYK?

There being n molecules, the total effect will be — -, — . The

entire inner surface of the cubical vessel being 6 P, the pressure p,
on unit area, will be

~ 6P ""3 I* '

but since P is the volume of the cube, which we will denote by v,
we have

1 mnc*
P = 3— '
or

pv = ^ • wnc2.

This is the fundamental equation of the kinetic theory of gases.
While the equation has been derived for a cubical vessel, it is
equally applicable to a vessel of any shape whatever, since the
total volume may be considered to be made up of a large number
of infinitesimally small cubes, for each of which the equation holds.
Deductions from the Kinetic Equation. Law of Boyle. In
the fundamental kinetic equation, pv = | mnc2, the right-hand
side is composed of factors which are constant at constant temper-
ature, and therefore the product, pv, must be constant also under
similar conditions. This is clearly Boyle's law.

GfASES 79

Law of Gay-Lussac. The kinetic equation may be written in
the form

pv = 3 ' 2 mmj2'

The kinetic energy of a single molecule being represented by
1/2 me2, the total kinetic energy of the molecules of gas will
be 1/2 mnc2. Therefore the product of the pressure and volume of
the gas is equivalent to two-thirds of the kinetic energy of its molecules.
A corollary to this proposition is that at constant pressure, the
average kinetic energy of the molecules in equal volumes of different
gases is the same. The law of Gay-Lussac teaches that at constant
volume, the pressure of a gas is directly proportional to its abso-
lute temperature. Taking this together with the fact that the
pressure of a gas at constant volume is directly proportional to ,
the mean kinetic energy of its molecules, it follows that the mean
kinetic energy of the molecules of a gas is directly proportional to its
absolute temperature. Thus we see that the absolute temperature of
a gas is a measure of the mean kinetic energy of its molecules. This
deduction is partially based upon the experimentally-determined
law of Gay-Lussac. Having obtained a definition of temperature
in terms of kinetic energy, it is easy to derive Gay-Lussac's law
from the fundamental kinetic equation. Writing the equation in
the form

2 1

it is apparent that pv is directly proportional to the total kinetic
energy of the gas molecules, or in other words, is directly propor-
tional to its absolute temperature, which is the most general
statement of Gay-Lussac's law.

Hypothesis of Avogadro. If equal volumes of two different gases
are measured under the^same pressure, we will have

pv = 1/3 ftiWiCi2 = 1/3 n2W2C22, (1)

where n\ and n2, mi and w2, and c\ and c2 denote the number, mass
and velocity of the molecules in the two gases. If the gases are

80 THEORETICAL CHEMISTRY

measured at the same temperature, the molecules of each possess
the same mean kinetic energy, or

1/2 mid2 = 1/2W2C22. (2)

Dividing equation (1) by equation (2), we have

HI = tt2,

or under the same conditions of temperature and pressure equal
volumes of the two gases contain the same number of molecules.
This is the hypothesis of Avogadro.

Law of Graham. If the fundamental kinetic equation be solved
for c, we have

but v/mn = 1/d, where d is the density of the gas, and therefore
we may write

If the pressure remains constant it is evident that the mean veloc-
ities of the molecules of two gases are inversely proportional to
the square roots of their densities, a law which was first enunciated
by Graham in 1833 as the result of his experiments on gaseous
diffusion.

Mean Velocity of Translation of a Gaseous Molecule. By
substituting appropriate values for the various magnitudes in the
equation

it is possible to calculate the mean velocity of the molecules of any
gas. Thus, for the gram-molecule of hydrogen at 0° and 76 cm.
pressure, p = 76x13.59 = 1033.3 gr. per sq. cm. = 1033.3 X 980.6
dynes per sq. cm., v = 22,400 cc., and mn = 2.016 gr.
Substituting these values in the above equation we have,

= /3 X 1033.3 X 980.6 X 22,400 _ mm cm
V 2.016

GASES 81

Thus at 0° the molecule of hydrogen moves with a speed slightly
greater than one mile per second. This enormous speed is only
attained along the mean free path, the frequent collisions with
other molecules rendering the actual speed much less than that
calculated.

Equation of van der Waals. As has been pointed out in a
previous paragraph, the gas laws are merely limiting laws and
while they hold quite well up to pressures of about 2 atmospheres,
above this pressure the differences between the observed and cal-
culated values become steadily larger. In the case of hydrogen,
Natterer was the first to show that the product of pressure and
volume is invariably higher than it should be. A possible explana-
tion of this departure from the gas laws was offered by Budde,
who proposed that the volume, v, in the equation pv = RT, should
be corrected for the volume occupied by the molecules them-
selves. If this volume correction be denoted by 6, then the gas
equation becomes

p (v - 6) = RT,

where b is a constant for each gas. Budde calculated the value
of 6 for hydrogen and found it to remain constant for pressures
varying from 1000 to 2800 meters of mercury.

While Budde's modification of the gas equation is quite satis-
factory in the case of hydrogen, it fails when applied to other gases.
In general, the compressibility at low pressures is considerably
greater than can be accounted for by Boyle's law. The compressi-
bility reaches a minimum value, and then increases rapidly so that
pv passes through the value required by the law. This suggests
that there is some other correction to be applied in addition to
the volume correction introduced into the gas equation by Budde.
van der Waals pointed out in 1879, that in the deduction of Boyle's
law by means of the fundamental kinetic equation, the tacit
assumption is made that the molecules exert no mutual attraction.
While this assumption is undoubtedly justifiable when the gas is
subjected to a very low pressure, it no longer remains so when
the gas is strongly compressed. A little consideration will make
it apparent that wtien increased pressure is applied to a gas, the

S2 THEORETICAL CHEMISTRY

resulting volume will become less than that calculated, owing to
molecular attraction. In other words the molecular attraction
and the applied pressure act in the same direction and the gas
behaves as if it were subjected to a pressure greater than that
actually applied, van der Waals showed that this correction is
inversely proportional to the square of the volume, and since it
augments the applied pressure the expression p + a/v2 is sub-
stituted for p in the gas equation, a being the constant of molecular
attraction. The corrected equation then becomes

(p + «A2) (v - 6) = RT.

This is known as the equation of van der Waals. It is applicable
not only to strongly compressed gases, but also to liquids as well.
While it will be given detailed consideration in a subsequent chapter,
it may be of interest to point out at this time the satisfactory ex-
planation which it offers of the experimental results of Amagat,
to which we have already made reference, (page 72). When v is
large, both 6 and a/v2 become negligible, and van der Waals'
equation reduces to the simple gas equation, pv = RT. We may
predict, therefore, that any influence tending to, increase v will
cause the gas to approach more nearly to the ideal condition. This
is in accord with the results of Amagat's experiments, which show
that an increase of temperature at constant pressure, or a diminu-
tion of pressure at constant temperature, causes the gas to tend to
follow the simple gas laws. The equation also offers a satisfactory
explanation of the exceptional behavior of hydrogen when it is
subjected to pressure. As we have seen, pv for all gases, except
hydrogen, diminishes at first with increasing pressure, reaches a
minimum value, and then increases regularly. Since the volume
correction in van der Waals' equation acts in opposition to the
attraction correction, it is apparent that at low pressures the effect
of attraction preponderates, while at high pressures the volume
correction is relatively of more importance. At some intermediate
pressure the two corrections counterbalance each other, and it is
at this point that the gas follows Boyle's law strictly. The
exceptional behavior of hydrogen may be accounted for by making
the very plausible assumption that the attraction correction is

GASES 83

negligible at all pressures in comparison with the volume correc-
tion.

Vapor Density and Molecular Weight. As has been pointed out
in an earlier chapter, when a substance can be obtained in the gas-
eous state, the determination of its molecular weight resolves itself
into finding the mass of that volume of vapor which will occupy
22.4 liters at 0° and 76 cm. It is inconvenient to weigh a volume
of gas or vapor under standard conditions of temperature and
pressure, but by means of the gas laws the determination made
at any temperature and under any pressure can be reduced to
standard conditions. For example, suppose v cc. of gas are found
to weigh w grams at t° and p cm. pressure, then the weight in grams
of 22.4 liters or 22,400 cc. at 0° and 76 cm. will be given by the fol-
lowing proportion, in which M denotes the molecular weight of the
substance: —

pv _ , , 76 X 22,400
Wl ~M: 273 '

^ X 76 X 22,400 X (t + 273)

, , 273 pv

1^ (x > i4 ^ 1+6C

The ^termination of vapor density may be effected in either of
two ways; (1) we may determine the mass of a known volume of
vapor under definite conditions of temperature and pressure, or
(2) we may determine the volume of a known mass under definite
conditions of temperature and pressure. There are a variety of
methods for the determination of vapor density; but for our pur-
pose it will be necessary to describe but two of them. In the
method of Regnault the mass of a definite volume of vapor is
determined, while in the method due to Victor Meyer we measure
the volume of a known mass.

Method of Regnault. In this method which is especially adapted
to permanent gases, use is made of two spherical glass bulbs
(Fig. 16) of approximately the same capacity, each bulb being
provided with a well-ground stop-cock. By means of an airpump
one bulb is evacuated as completely as possible, and is then filled,
at definite temperature and pressure, with the gas whose density

84 THEORETICAL CHEMISTRY

is~to be determined. The stop-cock is then closed and the bulb
weighed, the second bulb being used as a counterpoise. The use
of the second bulb is largely to avoid the
troublesome corrections for air displacement
and for moisture, each bulb being affected in
the same way and to nearly the same extent.
The volume of the bulb may be obtained by
weighing it first evacuated, and then filled with
distilled water at known temperature. From
these results we may calculate the mass per unit
of volume; or we may substitute the values of
w, v, p and t in the above formula and calcu-
late M, the molecular weight. This method
was used by Morley * in his epoch-making re-
search on the densities of hydrogen and oxygen.
Method of Victor Meyer, In the method
of Victor Meyer, a weighed amount of the lg*

substance is vaporized, and the volume which it would have
occupied at the temperature of the room and under existing
barometric pressure is determined. The apparatus of Meyer,
shown in Fig. 17, consists of an inner glass tube A, about 1
cm. in diameter and 75 cm. in length. This tube is expanded
into a bulb at the lower end, while at the top it is slightly en-
larged and is furnished with two side tubes C and E. The tube
A is suspended inside a heating jacket J3, containing some liquid
the boiling point of which is about 20° higher than the vaporizing
temperature of the substance whose vapor density is to be de-
termined. The side tube E dips beneath the surface of water in a
pneumatic trough G, and serves to convey the air displaced from
A to the eudiometer F. By means of the side tube C, and the glass-
rod Dy the small bulb containing the substance can be dropped to
the bottom of A. To carry out a determination of vapor density
with this apparatus, the liquid in B is heated to boiling and
the sealed bulb V, containing a weighed amount of the substance,
is placed in position on the rod D, the corks being tightly inserted.

* Smithsonian Contributions to Knowledge, (1895).

GASES

When bubbles of air cease to issue from E in the pneumatic trough,
showing that the temperature within A is constant, the eudiometer
F, full of water, is placed over the mouth of E, and the bulb V is
allowed to drop by drawing aside the rod D. Air bubbles immedi-
ately begin to issue from E and to collect in the eudiometer. When
the air ceases to collect, the eudiometer is closed by the thumb and

Fig. 17.

is removed to a large cylinder of water where it is allowed to stand
long enough to acquire the temperature of the room. It is then
raised or lowered until the level of water inside and outside is
the same, when the volume of air is carefully read off. In this
method, the substance on vaporizing displaces an equal volume
of air which is collected and measured, this observed volume being

86 THEORETICAL CHEMISTRY

that which the vapor would occupy after reduction to the condi-
tions under which the air is measured. It is evident that in this
method we do not require a knowledge of the temperature at
which the substance vaporizes. Since the air is measured over
water, the pressure to which it is subjected is that of the atmos-
phere diminished by the vapor pressure of water at the temperature
of the experiment. The method of calculating molecular weights
from the observations recorded may be illustrated by the follow-
ing example: — 0.1 gram of benzene (C6H6) was weighed out, and
when vaporized, 32 cc. of air were collected over water at 17° and
750 mm. pressure. The vapor pressure of water at 17° is 14.4
mm., and the actual pressure exerted by the gas is 750 — 14.4 =
735.6 mm. Substituting in the proportion

pv .. 760X22,400

":« + 273-M: 273 '

and solving for M we have

M - 0-1 X 760 X 22,400 X (17 + 273)

M ~ 273 X 735.6 X 32 "" 'b'*'

The result agrees fairly well with the molecular weight of benzene
(78.05) calculated from the formula.

Unless a vapor follows the gas laws very closely, the value of the
molecular weight obtained by the method of Victor Meyer will be
only approximate, but this approximate value will be sufficiently
near to the true molecular weight to enable us to choose between
the simple formula weight, given by chemical analysis, and some
multiple of it.

Results of Vapor-Density Determinations. As the result of
numerous vapor-density determinations extending over a wide
range of temperatures, much important data has been collected
concerning the number of atoms contained in the molecules of a
large number of chemical compounds. The molecular weights
of most of the elementary gases are double their atomic weights,
showing that their molecules are diatomic. In like manner the
molecular weights of mercury, zinc, cadmium and, in fact, all of
the vaporizable metallic elements have been found to be identi-
cal with their atomic weights. The molecules of sulphur,

GASES 87

arsenic, phosphorus and iodine are polyatomic, if they are not
heated to too high a temperature. The investigations of Meyer
and others have shown that the vapor densities of a large number
of substances diminish as the temperature is increased. In other
words as the temperature is raised the number of atoms contained
in the molecules decreases. The molecular weight of sulphur, cal-
culated from its vapor density at temperatures below 500°, corre-
sponds to the formula S8. If the vapor of sulphur is heated to
1100°, the molecular weight corresponds to the formula S2. In
fact, sulphur in the form of vapor may be represented by the formu-
las Sg, fi»4, $2, or even S according to the temperature at which its
vapor density is determined. Iodine behaves similarly, the mole-
cules being diatomic between 200° and 600°, while at temperatures
above 1400° the vapor density has about one-half its value at the
lower temperature, showing a complete breaking down of the dia-
tomic molecules into single atoms. Heating to yet higher tem-
peratures has failed to reveal any further decomposition. This
phenomenon is not confined to the molecules of the elements alone,
but is also met with in the case of the molecules of chemical com-
pounds. The vapor density of arsenious oxide between 500° and
700° corresponds to the formula As406. As the temperature is
raised, the vapor density becomes steadily smaller until, at 1800°, the
calculated molecular weight corresponds to the formula As203. In
like manner ferric and aluminium chlorides have been shown to
have molecular weights at low temperatures corresponding to the
formulas, Fe2Cl6 and A12C16. The commonly-used formulas, FeCk
and A1C13, represent their molecular weights at high temperatures
only. The experimental difficulties attending vapor density de-
terminations increase as the temperature is raised, owing chiefly to
the deformation of the apparatus when the material of which it is
constructed approaches its melting point. Glass which can be used
at relatively low temperatures only, has been replaced by specially
resistant varieties of porcelain which may be used up to tempera-
tures of 1500° or 1600°. Platinum vessels retain their shape up to
temperatures between 1700° and 1800°. Measurements up to
2000° have recently been effected by Nernst and his pupils.* In
* Wartenberg. Zeit. anorg. Chem., 56, 320 (1907).

88 THEORETICAL CHEMISTRY

their experiments use was made of a vessel of iridium, the inside
and outside of which was surrounded with a cement of magnesia
and magnesium chloride, the entire apparatus being heated electri-
cally. With this apparatus they showed that the molecular
weight of sulphur between 1800° and 2000° is 48, indicating that
the diatomic molecule is approximately 50 per cent broken down
into single atoms.

Abnormal Vapor Densities. In all of the cases cited above
the molecular weight calculated from the vapor density corre-
sponds either with the simple formula weight, as determined by
chemical analysis, or with a multiple thereof. In no case is there
any evidence of a breaking down of the simple molecule into its
constituents. Substances are known, however, the molecular
weights of which, calculated from their vapor densities, are less
than the sum of the atomic weights of their constituents. For
example, tne vapor density of ammonium chloride was found to
be 0.89, while that corresponding to the formula NH4C1 should be
1.89. Similar results have been obtained with phosphorus penta-
chloride, nitrogen peroxide, chloral hydrate and numerous other
substances. The phenomenon can be explained in either of the two
following ways: (1) that the molecule has undergone a complete
disruption, or (2) that the substance does not follow the law of
Avogadro. Until the former explanation was shown to be correct,
the latter was accepted and for a time the law of Avogadro fell into
disrepute. In 1857, Deville showed that numerous chemical com-
pounds are broken down or " dissociated " at high temperatures.
Shortly afterward Kopp suggested that the abnormal vapor
densities of such substances as ammonium chloride, phosphorus
pentachloride, etc., might be due to thermal dissociation. If
ammonium chloride underwent complete dissociation, one molecule
of the salt would yield one molecule of ammonia and one molecule
of hydrochloric acid gas, and the vapor density of the resulting
mixture would be one-half of that of the undissociated substance,
a deduction in complete agreement with the results of experiment.
It remained to prove that the products of this supposed dissocia-
tion were actually present.

The first to offer an experimental demonstration of the simul-

GASES

taneous formation of ammonia and hydrochloric acid, when ammon-
ium chloride is heated, was PebaL* The apparatus which he de-
vised for this purpose is shown in Fig. 18. It consisted of two tubes
T and t, the latter being placed within the former as indicated in
the sketch. Near the top of the inner tube, which was drawn down
to a smaller diameter, was a porous plug of asbestos, C, upon which
was placed a little ammonium chloride. A stream of dry hydro-

Rydrogen

Hydrogen

Fig. 18

gen was passed through the apparatus by means of the tubes A and
B, the former entering the outer tube and the latter the inner
tube. The entire apparatus was heated to a temperature above
that necessary to vaporize the ammonium chloride. If the salt
undergoes dissociation into ammonia and hydrochloric acid, the
former being less dense than the latter, would diffuse more
rapidly through the plug C and the vapor below the plug would

* Lieb,, Ann., 123, 199 (1862).

THEORETICAL CHEMISTRY

be relatively richer in ammonia than the vapor above it. The
current of hydrogen through jB would therefore sweep out from
the lower part of t an excess of ammonia, while the current through
A would carry out from T an excess of hydrochloric acid. By
holding strips of moistened litmus paper in the currents of gas
issuing from E and F, it was possible for Pebal to test the correct-
ness of Kopp's idea. He found that the gas issuing from E had
an acid reaction while that escaping from F had an alkaline reac-
tion. It would at first sight appear that Pebal had demonstrated

Nitrogen

Fig. 19.

beyond question that ammonium chloride undergoes dissociation
into ammonia and hydrochloric acid.

It was pointed out, however,*that Pebal had heated the ammon-
ium chloride in contact with a foreign substance, asbestos, and
that this might have acted as a catalyst, promoting the decomposi-
tion into ammonia and hydrochloric acid. This objection was
removed by the ingenious experiment of Than.* He devised a
modification of Pebal's apparatus, as shown in Fig. 19. In the
horizontal tube, AB, the ammonium chloride was placed at F and a

* Lieb. Ann., 131, 129 (1864).

GASES 91

porous plug of compressed ammonium chloride was introduced at
G. The tube was heated and nitrogen passed in at C. The
reactions of the currents of gas issuing at D and E were tested
with litmus as in Pebal's experiment and it was found that the
gas escaping from D was alkaline, while that issuing from E was
acid. This experiment proved beyond question that the vapor
of ammonium chloride is thermally dissociated into ammonia
and hydrochloric acid. Experiments on other substances whose
vapor densities are abnormally small show that a similar explan-
ation is applicable, and thus furnish a confirmation of the law of
Avogadro.

Calculation of the Degree of Dissociation. Since the density
of a dissociating vapor decreases with increase in temperature,
it is important to be able to calculate the degree of dissociation at
any one temperature. This is clearly equivalent to ascertaining
the extent to which the reaction

has proceeded from left to right. This can be determined easily
from the relation of vapor density to dissociation. If we start
with one molecule of gas and let a represent the percentage dis-
sociation, then 1 — a will denote the percentage remaining un-
dissociated. If one molecule of gas yields n molecules of gaseous
products, the total number of molecules present at any time will
be

(1 - a) + na = 1 + (n - 1) a.

The ratio 1 : 1 + (n — 1) a will be the same as the ratio of the
density cfe of the dissociated gas to its density in the undissociated
state do, or

1 : 1 + (n - 1) a = 4 : A;

solving this proportion for a, we have

a =

(n - 1) d*

The vapor density of nitrogen peroxide has been measured by E.

and L. Natanson,* and the degree of dissociation at the different

* Wied. Ann., 24, 454 (1885); 27, 606 (1886).

THEORETICAL CHEMISTRY

temperatures calculated by means of the preceding formula. The
following table gives their results.

The course of the dissociation is shown in the accompanying
illustration, Fig. 20, in which the abscissae represent temperature
and the ordinates, percentage dissociation. It will be observed

80 100

Temperature

Fig. 20.

that the dissociation of nitrogen peroxide is at first nearly pro-
portional to the temperature. It then increases more rapidly
until, when about four-fifths of the molecules of N204 are broken
down, the dissociation proceeds slowly to completion.

Specific Heat. The addition of heat energy to a body causes
its temperature to rise. The ratio of the amount of heat supplied
to the resulting rise in temperature is called the heat capacity of
the body; obviously its value is dependent upon the initial temper-

GASES

DISSOCIATION OF NITROGEN PEROXIDE, N2O4.

ATMOSPHERIC PRESSURE.
(Density of N2O4=3.18; of NO2+NO2 = 1.59; of air = 1.00.)

Temperature,
(degrees)

Density of Gas.

Percentage Dis-
sociation.

26.7

2.65

19 96

35.4

2 53

25 65

39 8

2 46

29 23

49 6

2 27

40 04

60.2

2 08

52 84

70 0

1 92

65 57

80 6

1 80

76 61

90 0

1 72

84 83

100 1

1 68

89 23

111 3

1 65

92 67

121 5

1 62

96 23

135 0

1.60

98 69

154 0

1 58

100 00

ature of the body. The specific heat of a substance may be defined
as the heat capacity of unit mass of the substance. If dt represents
the increment of temperature due to the addition of dQ units of
heat energy to m grams of any substance, then its specific heat, c,
will be given by the equation

..
m dt

Specific Heat at Constant Pressure and Constant Volume. It

is well known that the specific heat of a gas depends upon the
conditions under which it is determined. If a definite mass of
gas is heated under constant pressure, the value of the specific
heat, cp, is different from the value of the specific heat, cv, ob-
tained when the pressure varies and the volume remains con-
stant. The value of cp is invariably greater than that of cv.
When heat is supplied to a gas at constant pressure not only does
its temperature rise, but it also expands, and thus does external
work. On the other hand, if the gas be heated in such a way that
its volume cannot change, none of the heat supplied will be used
in doing external work, and consequently its heat capacity will

94 THEORETICAL CHEMISTRY

be less than when it is heated under constant pressure. The
recognition by Mayer in 1841 of the cause of this difference between
the two specific heats of a gas led him to his celebrated calculation
of the mechanical equivalent of heat, and the enunciation of the
first law of thermodynamics. Mayer observed that the differ-
ence between the quantity of heat necessary to raise the temper-
ature of 1 gram of air 1° C. at constant pressure, and at constant
volume respectively, was 0.0692 calorie, or
cp - cv = 0.0692 cal.

That is to say, 0.0692 calorie is the amount of heat energy which
is equivalent to the work Squired to expand 1 gram of air 1/273
of its volume at 0°. Imagine 1 gram of air at 0° enclosed within
a cylinder having a cross-section of one square centimeter, and
furnished with a movable, frictionless piston. Since 1 gram of
air under standard conditions of temperature and pressure occu-
pies 773.3 cc., the distance between the piston and the bottom of
the cylinder will be 773.3 cm. If the temperature be raised from
0° to 1°, the piston will rise 1/273 X 773.3 = 2.83 cm., and since
the pressure of the atmosphere is 1033.3 grams per square centi-
meter, the external work done by the expanding gas will be

1033.3 X 2.83 = 2924.3 gr. cm.

This is evidently equivalent to 0.0692 calorie and therefore, the
equivalent of 1 calorie in mechanical units, J, will be

0004. Q

a value agreeing very well with the best recent determinations of
the mechanical equivalent of heat.

The difference between the two specific heats may be easily
calculated in calories from the fundamental gas equation. Start-
ing with 1 mol. of gas, and remembering that when a gas expands
at constant pressure, the product of pressure and change in volume
is a measure of the work done, we have, at temperature TV0,

where Vi is the molecular volume. Raising the temperature to

GASES 95

T2°, the corresponding molecular volume being t;2, we have for
the work done during expansion

If r2 — TI = 1°, then the equation reduces to

p (^2 - vi) = R.

Since the difference between the molecular heats * at constant
pressure and constant volume is equivalent to the external work
involved when the temperature of 1 mol. of gas is raised 1°, we have

M (cp - cv) = p (v2 - t>i),

where M is the molecular weight of the gas; and therefore
M (cp — cv) = R = 2 calories.

In words, the difference of the molecular heats of any gas at
constant pressure and at constant volume is 2 calories. The
specific heat of a gas at constant pressure can be readily deter-
mined, by passing a definite volume of the gas, heated under con-
stant pressure to a known temperature, through the worm of a
calorimeter at such a rate that a constant difference is maintained
between the temperature of the entering and the temperature of
the escaping gas. Thus the number of calories which causes a
definite thermal change in a certain volume of the gas is deter-
mined, and from this it is an easy matter to calculate the specific
heat, cp. The molecular heat at constant pressure for all gases
approaches the limiting value, 6.5, at the absolute zero. This
relation, due to Le Chateiier, may be expressed thus,

Mcp = 6.5 + aT,

where a is a constant for each gas. The value of a tor hydrogen,
oxygen, nitrogen and carbon monoxide is 0.001, for ammonia,
0.0071 and for carbon dioxide, 0.0084. As the complexity of the
gas increases the value of a becomes numerically greater.

The experimental determination of the specific heat of a gas at
constant volume is difficult and the results obtained are not
trustworthy. The chief cause of the inaccuracy of the results

* The molecular heat of a gas is equal to the product of its specific heat and its
molecular weight.

THEORETICAL CHEMISTRY

is that the vessel containing the gas absorbs so much more heat
than the gas itself that the correction is many times larger than
the quantity to be measured. The specific heat at constant vol-
ume is almost always obtained by indirect methods, as for example
by means of the preceding formula

M (cp - cv) = R = 2 cal.,

in which the values of M and cp are known.

The molecular heats of some of the commoner gases and vapors
are given in the subjoined table together with the ratio cp/cv.

MOLECULAR SPECIFIC HEATS.

Gas.

Mcp

Me.

<>/<•„= 7

Argon

1 66

Helium

1 66

Mercury

1 66

Hydrogen

6 88

4 88

1 41

Oxygen

6 96

4 96

1 40

Nitrogen

6 93

4 93

1 41

Chlorine

8 58

6,58

1 30

Bromine

8 88

6 88

1 29

Nitric oxide

6 95

4 95

1 40

Carbon monoxide

6 86

4 86

1 41

Hydrochloric acid

6 68

4 68

1.43

Carbon dioxide

9 55

7 55

1 26

Nitrous oxide
Water

9 94
8 65

7 94
6 65

1 25

1 28

Sulphur dioxide

9 88

7.88

1 25

Ozone

1 29

Ether ...

35 51

33.51

1 06

The Ratio of the Two Specific Heats. There are two methods
by which the ratio cp/cv can be determined directly, one due to
Clement and Desorrnes * and the other due to Kundt.f

Method of Clement and Desormes. The apparatus devised by
these investigators consists, as is shown in Fig. 21, of a glass
balloon flask, A, of about 20 liters capacity, furnished with two
stop-cocks, D and 13, and a manometer, C. The stop-cock D
has an aperture nearly as large as the diameter of the neck of the

* Jour, de phys., 89, 321, 428 (1819).

t Pogg. Ann., 128, 497 (1866); 135, 337, 527 (1868).

GASES

flask, B. To determine the ratio of the two specific heats, the
flask is filled with the gas under a pressure slightly greater than
barometric pressure. The manometer C serves to measure the
pressure of the gas within A. After the value of the pressure
has been read on the manometer, the stop-cock D is opened
momentarily to the air, thus permitting the pressure of the gas
to fall adiabatically to that of the atmosphere. The stop-cock

Fig. 21.

is then closed and the flask is allowed to stand for a few moments
until its contents, which has cooled by adiabatic expansion, has
regained the temperature of the room. The pressure on the
manometer is then observed. Let the initial pressure of the gas
be denoted by pQ, and atmospheric pressure by P. If the initial
and final specific volumes are denoted by VQ and v\, then for an
adiabatic process, we have

P = /voV

Po W '

98 THEORETICAL CHEMISTRY

The value of the final specific volume is determined from the
final pressure, pi, by an application of Boyle's law, the pressure pi
being developed isothermally.
Thus,

VjO^Pl^
Vi po'

and consequently,

= logP- logpo
log pi- log po*

Method of Kundt. According to the formula of Laplace for the
velocity of transmission of a sound wave in a gas, we have

in which p and d denote the pressure and density of the gas, and
7 is the ratio of the two specific heats. If the wave velocities in
two different gases, whose densities are d\ and cfe under the same
conditions of temperature and pressure, be denoted by v\ and vz,
we may write

or replacing the densities of these gases by their respective molec-
ular weights, MI and M2, we have

TI/T

The ratio of the velocities of the two waves can be measured by
means of the apparatus shown in Fig. 22. A wide glass tube
about \\ meters in length is furnished with two side tubes, E and
F. Into one end of the tube is inserted the glass rod BD which
is clamped at its middle point by a tightly fitting cork, C. The
other end of the tube is closed by means of the plunger A. A
small amount of lycopodium powder is placed upon the bottom of

GASES 99

the tube and is distributed uniformly by gently tapping the walls
of the tube. The gas in which the velocity of the sound wave
is to be determined is introduced into the tube through JB, and
the displaced air escapes at F. When the tube is filled, E and F
are closed by means of rubber caps, and a piece of moistened
chamois leather is drawn along BD causing it to vibrate longitudi-
nally and to emit a shrill note. The vibrations are taken up by the

Fig. 22.

gas in the tube and the powder arranges itself in a series of heaps
corresponding to the nodes of vibration. If the nodes are not
sharply defined, then A should be moved in or out until they
become so. If Xi is the distance between two heaps or nodes,
then 2 Xi will be the wave length of the note emitted by the rod
BD, and if n represents the number of vibrations per second of
the note emitted, we have for the velocity of sound in the gas

Similarly if a second gas be introduced into the tube we shall
have

Therefore,

? - r

vt X2
Substituting in equation (1), we have

If the second gas is air, as is usually the case, 72 = 1.405 and M2 =
28.74, (mol. wt. of hydrogen -5- density of hydrogen referred to air,
or 2 •*• 0.0696 = 28,74) or equation (3) becomes

100 THEORETICAL CHEMISTRY

1 A f\ K Xl -M- 1

^ = L405V>2^7i-

Thus, 7 for any gas can be determined by this method provided
we know its value for another gas of known molecular weight.

Specific Heat of Gases and the Kinetic Theory. In terms of
the kinetic theory, the energy of a gas may be considered to be
made up of three parts: (1) the translatioiial energy of the mole-
cules, commonly termed their kinetic energy, (2) the intramolec-
ular kinetic energy, and (3) the potential energy due to inter-
atomic attraction within the molecules. When a gas is heated
at constant volume all three of these factors of the total energy
of the molecule may be affected. It is fair to assume, however,
that when a monatomic gas, such as mercury vapor, is heated,
all of the heat energy supplied is used to augment the translational
kinetic energy of the molecules. As we have seen, the fundamental

kinetic equation

pv = 1/3 nmc2
may be written

pv = 2/3 -1/2 nmc2,

and since 1/2 nmc? represents the total kinetic energy of the gas,

we have

pv = 2/3 kinetic energy of 1 mol.,
or

kinetic energy of 1 mol. = 3/2 pv.

But pv = 2 T calories, therefore

kinetic energy of 1 mol. = 3 T cal.

The kinetic energy of a constant volume of any gas at the temper-
tures TI and T2, is given by the following equations: —

3/2 2W = 3T7!, (1)

and

3/2 p2v = 3 !F2. (2)

Subtracting (1) from (2) we obtain

3/2 (ft - pi) t> = 3 (T2 - TO, (3)

and for an increase in temperature of 1°, (3) becomes
3/2 (p2 - Pi) v = 3 cal.

GASES 101

The molecular kinetic energy of one mole of a monatomic gas
at constant volume is thus increased by 3 calories for each degree
rise in temperature. As has already been shown,

M (cp - c,) = 2 cal,
therefore, since Mcv = 3 calories, Mcp = 3 + 2 = 5 calories, and

/t/ r ^\
*, — 1KiCP — ~ — 1 Afi

7 ~ Mcv ~ 3 " Lbb*

This value of 7 is in perfect agreement with the results of the
experiments on mercury vapor which is known to be monatomic.
The converse of this method has been employed by Ramsay to
prove that the rare gases of the atmosphere are monatomic, the
value of 7 for all of these gases being 1.66. In the case of poly-
atomic molecules the heat energy supplied is not only used in
increasing their translational kinetic energy, but also in the
performance of work within the molecule. The value of the
internal work is indeterminate, but it is without doubt constant
for any one gas. If the internal work be represented by a, then
the value of the ratio of the two specific heats will be

7 = ^ = |+|< UK >l.

Reference to the table on p. 96, giving the value of 7 for differ-
ent gases, will show that this deduction from the kinetic theory
is in perfect agreement with the experimental facts. With
increasing complexity of the molecule, it is apparent that the
amount of heat expended in doing internal work should increase,
and therefore the specific heat should increase also. Inspection
of the table confirms this deduction. The specific heat of mona-
tomic gases is independent of the temperature while the specific
heat of polyatomic gases increases slightly. These results may
justly be regarded as among the greatest triumphs of the kinetic
theory of gases.

102 THEORETICAL CHEMISTRY

PROBLEMS.

1. The volume of a quantity of gas is measured when the barometer
stands at 72 cm., and is found to be 646 cc.: what would its volume be
at normal pressure? Ans. 612 cc.

2. At what pressure would the gas in the preceding problem have a
volume of 580 cc.? Ans. 80.19 cm.

3. A certain quantity of oxygen occupies a volume of 300 cc. at 0°:
find its volume at 91°. Ans. 400 cc.

4. The weight of a liter of air under standard conditions is 1.293 grams:
to what temperature must the air be heated so that it may weigh exactly
1 gram per liter? Ans. 79°.99.

5. At what temperature will the volume of a given mass of gas be
exactly double what it is at 17°? Ans. 307°.

6. On heating a certain quantity of mercuric oxjde it is found to give
off 380 cc. of oxygen, the temperature being^Fjand the barometric
height 74 cm.; what would be the volume of tHe gas under standard
conditions? Ans. 341.25 cc.

7. A liter of air weighs 1.293 grains under standard conditions. At
what temperature will a liter of air weigh 1 gram, the pressure being 72 cm.?

Ans. 61°.43.

8. A quantity of air at atmospheric pressure and at a temperature of
7° is compressed until its volume is reduced to one-seventh, the temper-
ature rising 20° during the process: find the pressure at the end of the
operation. Ans. 7.3 atmos.

9. The weight of a liter of nitrogen under standard conditions is 1.2579
grams. Calculate the specific gas constant, r. Ans. 3007 gr. cm.

10. The time of outflow of a gas is 21.4 minutes, the corresponding
time for hydrogen is 5.6 minutes. Find the molecular weight of the gas.

Ans. 29.2.

11. Calculate the molecular weight of chloroform from the following
data: —

Weight of chloroform taken 0.220 gr.

Volume of air collected over water 45.0 cc.

Temperature of air 20°

Barometric pressure 755.0 mm.

Pressure of aqueous vapor at 20° 17.4 mm.

Ans. 121.1.

GASES 103

12. The density of a gas is 0.23 referred to mercury vapor. What is
its molecular weight? Ans. 46.

13. Phosphorus pentachloride dissociates according to the equation

The molecular weight PCl« is 208.28. At 182° the density is 73.5 and
at 203° it is 62. Find the degree of dissociation at the two temperatures.

Ans. ai82° = 0.417, oW = 0.68.

14. The specific heat at constant volume for argon is 0.075, and its
molecular weight is 40. How many atoms are there in the molecule?

Ans. 1.

15. What is the specific heat of carbon dioxide at constant volume, its
molecular weight being 44 and the temperature 50°. Ans. 0.164.

16. The specific heat for constant pressure of benzene is 0.295: what is
the specific heat for constant volume? Ans. 0.27,

CHAPTER VII.
LIQUIDS.

General Characteristics of Liquids. The most marked char-
acteristic of the liquid state is that a given mass of liquid has a
definite volume but no definite form. The volume of a liquid is
dependent upon temperature and pressure but to a much smaller
degree than is the volume of a gas. The formulas in which the vol-
ume of a liquid is expressed as a function of temperature and pres-
sure are largely empirical, and contain constants dependent upon
the nature of the liquid. This is undoubtedly due to the fact that
in the liquid state the molecules are much less mobile than in the
gaseous state. The distance between contiguous molecules being
much less in liquids than in gases, the mutual attraction is increased
while the mobility is correspondingly diminished. That liquids
represent a more condensed form of matter than gases is shown
by the change in volume which results when a liquid is vaporized:
thus, 1 cc. of water at the boiling point when vaporized at the same
temperature occupies a volume of about 1700 cc. A liquid con-
tains less energy than a gas, since energy is always required to
transform it into the gaseous state. Since gases can be liquefied
by increasing the pressure and lowering the temperature, and since
liquids can be vaporized by lowering the pressure and increasing
the temperature, it is apparent that there is no generic difference
between the two states of matter.

Connection Between the Gaseous and Liquid States. If a gas
is compressed isothermally, its state may change in either of two
ways depending upon the temperature: — (1) The volume at first
diminishes more rapidly than the pressure increases, then in the
same ratio and lastly more slowly. When the pressure attains a
very high value the volume is but slightly altered. This case
has already been considered in the preceding chapter. (2) The
volume changes more rapidly than the pressure until, when a cer-

104

LIQUIDS 105

tain pressure is reached, the gas ceases to be homogeneous, partial
liquefaction resulting. For a constant temperature, the pressure at
which liquefaction occurs is invariable for a given gas, while the vol-
ume steadily diminishes until liquefaction is complete. Only when
the whole mass of gas has been liquefied is it possible to increase
the pressure and then, owing to the small compressibility of liquids,
a large increase in pressure is required to produce a slight dimin-
ution in volume. If the temperature is above a certain point,
dependent upon the nature of the gas, the phenomena of com-
pression will follow (1) ; if below this point, the process will follow
(2). That a gas may behave in either of the above ways was
first clearly recognized by Andrews* in 1869, in connection with
his experiments on the liquefaction of carbon dioxide. He found
that if carbon dioxide was compressed, keeping the temperature
at 0°, the volume changes more rapidly than the pressure, lique-
faction resulting when a pressure of 35.4 atmospheres was reached.
As the temperature was raised, he found that a higher pressure
was required to liquefy the gas, until at temperatures above
30°.92 it was no longer possible to condense the gas to the liquid
state. The temperature above which it was no longer possible
to liquefy the gas he termed the critical temperature. In like
manner the pressure required to liquefy the gas at the critical
temperature, he termed the critical pressure, and the volume
occupied by the gas or the liquid under these conditions he called
the critical volume.

Isothermals of Carbon Dioxide. The results of Andrew's
experiments f on the liquefaction of carbon dioxide are shown in
Fig. 23, in which the ordinates represent pressures and the abscissae
the corresponding volumes at constant temperature. The curves
obtained by plotting volumes against pressures at constant
temperatures are called isothermals. For a gas which follows
Boyle's law, the isothermals will be a series of equilateral hy-
perbolas. This condition is approximately fulfilled by air, for
which three isothermals are given in the diagram. At 48°. 1 the
isothermal for carbon dioxide is nearly hyperbolic, but as the

* Trans. Roy. Soc. 159, 583 (1869).
t loc. cit.

106

THEORETICAL CHEMISTRY

temperature becomes lower, the isothermals deviate more and more
from those for an ideal gas. At the critical temperature, 30°.92,

Carbon Dioxide—

Air

Volume
Fig. 23.

the curve is almost horizontal for a short distance, showing that for
a very slight change in pressure there is an enormous shrinkage
in volume. At still lower temperatures, 21°.l and 13°. 1, the

LIQUIDS 107

horizontal portions of the curves are much more pronounced,
indicating that during liquefaction there is no change in pressure.
When liquefaction is complete the curves rise abruptly, showing
that the change in volume is extremely small for a large increase
in pressure; in other words the liquefied gas possesses a small
coefficient of compressibility. At any point within the parabolic
area, indicated by the dotted line ABC, both vapor and liquid are
coexistent; at any point outside, only one form of matter, either
liquid or vapor, is present. Andrew's experiment show that
there is no fundamental difference between a gas and a liquid.
It is apparent from the diagram that when carbon dioxide is sub-
jected to great pressures above its critical temperature it behaves
more like a liquid than a gas, in fact it is difficult to determine
whether a highly compressed gas above its critical temperature
should be classified as a gas or as a liquid.

Van der Waals' Equation and the Continuity of the Gaseous
and Liquid States. In the preceding chapter we have learned
that the fundamental gas equation

pv = RT

is only strictly applicable to an ideal gas, and that the behavior
of actual gases is represented with considerable accuracy, even at
high pressures, by the equation of van der Waals,

If this equation be arranged in descending powers of v, we have

3 .A.. RT\ . a ab n n\

z>3 — v2 [ b H -- ) + v --- = 0. (1)

\PIPP

This being a cubic equation has three possible solutions, each
value of p affording three corresponding values of v; a, 6, R and T
being treated as constants. The three roots of this equation are
either all real, or one is real and two are imaginary, depending upon
the values of the constants. That is to say, at one temperature
and pressure the values of a and b may be such, that v has three
real values, while at another temperature and pressure, v may
have one real and two imaginary values. In the accompanying

108

THEORETICAL CHEMISTRY

diagram, Fig. 2$II#, series of graphs of the equation for different
values of T is given. It will be observed that these curves bear
a striking resemblance to the isotherms of carbon dioxide estab-
lished by the experiments of Andrews. In the case of the theo-

Volume
Fig. 24.

retical curves there are no sudden breaks such as appear in the
actual discontinuous passage from the gaseous to the liquid state.
Instead of passing from B to D along the wavelike path BaCbD,
experiment has shown that the substance passes directly from the
state B to the state D along the straight line BD. It is here

LIQUIDS 109

that van der Waals' equation fails to apply. As has been pointed
out the substance between these two points is not homogeneous,
being partly gaseous and partly liquid. Attempts have been
made to realize the portion of the curve BaCbD experimentally.
By studying supersaturated vapors and superheated liquids it
has been found possible to follow the theoretical curve for short
distances between B and D without discontinuity, but owing to
the instability of the substance in this region, it is evident that
the complete isothermal and continuous transformation of a gas
into a liquid can never be effected, van der Waals has called
attention to the fact that in the surface layer of a liquid, where
unique conditions prevail, it is quite possible that such unstable
states may exist, and that there the transition from liquid to gas
may in reality be a continuous process. The diagram shows that
as T increases, the wave-like portion of the isothermals becomes
less pronounced and eventually disappears, when the points jB,
C and D coalesce. At this point the three roots of the equa-
tion become equal, the volume of the liquid becoming identical
with the volume of the gas. The substance at this point is in the
critical condition. Since under these conditions the three roots
of the equation

3 L i RT\ 2 i a ab A m

v3 — { b H -- 1 v2 + - v -- =0 (1)

\ p / p p

are equal, we may write #1 = #2 = v3 = vc, the subscript c indicat-

ing the critical state. Then equation (1) must be equivalent to

(v - Vc)3 = v* - 3 Vjp + 3 Vfy - Vc* = o. (2)

Equating the corresponding coefficient of equations (1) and (2),
we have

8*-£ (4)

and vcz = — (5)

Dividing equation (5) by equation (4), we have

*>c = 36, (6)

110 THEORETICAL CHEMISTRY

and substituting this value in equation (4), we obtain

Lastly, substituting the values of vc and pc, given in equations
(6) and (7), in equation (3), we have

Tc = 8a (8)

Therefore,

"1R- (9)

Or expressing the constants a, b and R in terms of the critical
values of pressure, temperature and volume, we have

a = 3 pcv*, (10)

6 = \ (11)

and R = |^- (12)

By means of equations (6), (7) and (8) it is possible to calculate
the critical constants of a gas when the constants a and b of van
der Waals' equation are known. If we take carbon dioxide as an
example, for which a = 0.00874 and b = 0.0023 we obtain
vc = 0.0069 (observed value = 0.0066), pc = 61 atmospheres,
(observed value = 70 atmospheres), TV = 305°.5 abs. (observed
value 303°.9 abs.) Conversely by means of equations (10) and
(11) the value of a and 6 can be calculated when the critical data
are given.

Corresponding Conditions. If in the equation of van der
Waals, the values of p, v and T be expressed as fractions of the
corresponding critical values, we may write

V = 0VC

and

T = yTc.

LIQUIDS 111

Substituting these values in the equation

we have

(«p. + 7srH)G&«-*>-»yZI.>

and replacing pc, vc, and Tc by their values given in equations (6),
(7) and (8) of the preceding paragraph, we obtain

which is van der Waals' reduced equation of condition.

In this equation everything connected with the individual
nature of the substance has vanished, thus making it applicable
to all substances in the liquid or gaseous state in the same way
that the fundamental gas equation is applicable to all gases irre-
spective of their specific nature. It has been shown, however,
that the equation is not entirely trustworthy and at best can be
considered as little more than a rough approximation. The
chief point to be observed in connection with this equation is
that whereas for gases, the corresponding values of temperature,
pressure and volume, measured in the ordinary units, may be
compared, it is necessary in the case of liquids to make the com-
parison under corresponding conditions. For example, the molec-
ular volumes of two liquids are to be compared, not at room
temperature but at temperatures which are equal fractions of
their respective critical temperatures. Such temperatures van
der Waals called corresponding temperatures.

By way of illustration, suppose we wish to compare alcohol
and ether with respect to some particular property, such as
surface tension. If the surface tension of alcohol be measured
at 60°, at what temperature must a similar measurement be
made with ether in order that the results may be comparable?
The critical temperature of alcohol is 243° C. or 516° absolute;
that of ether is 194° C, or 467° absolute. Then according to van
der Waals' definition of corresponding conditions, the tempei^ture,

112

THEORETICAL CHEMISTRY

t, at which measurements should be made with ether will be given
by the proportion, 273 + t : 467 :: 273 + 60 : 516, or t = 28° C.
By making comparisons of various properties at corresponding
temperatures it has been found that greater regularities are
observed than when comparisons are made at the same tempera-
ture, thus justifying the claim of van der Waals.

Liquefaction of Gases. The history of the liquefaction of
gases has for a long time been regarded as one of the most
interesting chapters of physical science. Among the first success*

Kg. 25.

ful workers in this field was Faraday.* He liquefied practically
all of the gases which condense under moderate pressures and at
not very low temperatures. A sketch of the apparatus used by
Faraday is shown in Fig. 25. It consisted of an inverted
V-shaped tube, in one end of which was placed some solid which
would liberate the desired gas on heating, while the other end
was sealed and immersed in a freezing mixture. When the sub-
stance at A had been heated long enough to liberate considerable
gas, the pressure within the tube became sufficiently high to cause
the gas to liquefy at the temperature of the end B. Thus chlorine

* PhU. Trans., 113, 189 (1823).

LIQUIDS 113

hydrate was heated in the tube and the liberated chlorine was
condensed at B as a yellow liquid. IjnJ^^jJQiilQr^^
in liquefying carbon dioxide in quite large amounts by the use of
a new form of apparatus. In connection with his experiments
ori liquidTcarbon dioxide, he observed that when it was allowed to
vaporize, enough heat was absorbed to lower the temperature
below its freezing point, solid carbon dioxide being obtained. He
discovered that a mixture of solid carbon dioxide and ether was a
powerful refrigerant, and that under diminished pressure the
mixture g&ve temperatures ranging from — 100° C. to — 110° C.
This mixture is known today as Thilorier's mixture. Faraday f
undertook the liquefaction of the so-called permanent gases in 1845.
In this second series of experiments by Faraday he employed
higher pressures than in his earlier experiments, and also made
use of the newly discovered Thilorier mixture as a refrigerant.
He was partially successful in his attempt to liquefy the hitherto
noncondensible gases. He liquefied ethylene, phosphine and
hydrobromic acid and also solidified ammonia, cyanogen, and
nitrous oxide. He failed to liquefy hydrogen, oxygen, nitrogen,
nitric oxide and carbon monoxide. No further advance in the
liquefaction of gases was made until the year 1869 when Andrews
pointed out the importance of cooling the gas below its critical
temperature. This discovery explained why so many of the
earlie^ experiments had failed, and opened the way to the brilliant
successes of the latter part of the nineteenth century. In 1877,
Cailletet J and Pictet, § working independently, succeeded in
liquefying oxygen. Cailletet subjected the gas to a pressure of
about 300 atmospheres using boiling sulphur dioxide as a refriger-
ant. The gas was further cooled by suddenly releasing the pres-
sure and allowing it to expand, In addition to oxygen he also
liquefied air, nitrogen and possibly hydrogen. Shortly afterward
in 1883, the Polish scientists, Wroblewski and Olszewski, <[[ pub-

* Lieb. Ann., 30, 122 (1839).
t Phil. Trans., 135, 155 (1845).
J Compt. rend., 85, 1217 (1877).
§ Ibid., 85, 1214, 1220 (1877).
| Wied. Ann., 20, 243 (18&3).

114 THEORETICAL CHEMISTRY

lished an account of their interesting and highly important work.
In their experiments they subjected the gas to be liquefied to high
pressure, and simultaneously cooled it to a very low temperature.
Among the refrigerants used by them was liquid ethylene, which
was allowed to boil off under diminished pressure, giving a temper-
ature of — 130° C. At this temperature, a pressure of only
20 atmospheres was sufficient to condense oxygen to the liquid
state. Having liquefied oxygen, nitrogen, air and carbon mon-
oxide/ and having determined the boiling-points of these gases
under atmospheric pressure, they proceeded to use the&e liquefied
gases as refrigerants, allowing them to boil off undfcr diminished
pressure, thus obtaining temperatures as low as — 200° C. A
very small amount of liquid hydrogen was obtained in this way.
Subsequent attempts by these same experimenters to liquefy
hydrogen, while not much more successful than their former
attempts, enabled them to determine its boiling-point. Shortly
after the publication of the first papers of Wroblewski and
Ola&ewski, Dewar * devised a new form of apparatus for lique-
fying air, oxygen and nitrogen on a comparatively large scale.
He also introduced the well-known vacuum-jacketed flasks and
tubes which greatly facilitated carrying out experiments with
liquefied gases. In 1895, Linde in Germany and Hampson in
England simultaneously and independently constructed machines
for the liquefaction of air in large quantities.

In the method devised by these experimenters the air is not
subjected to a preliminary cooling, produced by the rapid evapora-
tion of a liquefied gas under diminished pressure, as in the methods
of Wroblewski and Olszewski. In the Linde liquefier, the air is
compressed to about 200 atmospheres. It is then passed through
a chamber containing anhydrous calcium chloride to remove the
greater part of the moisture, after which it is cooled by allowing it
to circulate through a coiled pipe immersed in a freezing mixture.
Nearly all of the moisture remaining in the air is deposited on the
walls of the pipe in the form of frost. The air then enters a long
spiral tube jacketed with a non-conducting material, and is there
allowed to expand to a pressure of about 15 atmospheres.
* Proc. Roy. Inst., 1886, 560.

LIQUIDS 115

During this expansion the temperature of the air is appreciably
lowered. When the air has traversed the spiral tube, it is still
further cooled by allowing it to expand to a pressure equal to that
of the atmosphere. The air which has been thus cooled is then
passed backward through the annular space between the spiral tube
and a concentric jacker, thus cooling the entering portion of air.
Consequently this next portion of air expands from a lower initial
temperature, and the cooling effect is increased. In like manner,
when this cooler air passes backward it cools still further the next
succeeding portion, and eventually the temperature is reduced
sufficiently to cause a small amount of the air to liquefy as it
issues from the end of the spiral tube. The remaining portion of
the air which has not been liquefied, passes backward through the
annular tube and cools the following portion to a still greater
extent, causing a larger proportion to liquefy on expansion. With
a 3-horse-power machine, a continuous supply of 0.9 liter per hour
can be obtained. Further improvements in this apparatus have
been made by Dewar and Hampson, and by means of it Dewar's
brilliant successes in the liquefaction of gases have been achieved.
The most efficient apparatus for the liquefaction of air and other
gases is that developed by Claude.* The essential features of
his liquefier are shown in Fig. 26. The air is first compressed to
40 atmospheres pressure by means of an ordinary compression
pump not shown in the diagram, the moisture and carbon dioxide
being removed as in Linde's method. It then enters the tube A,
which in reality is of a spiral form, and divides at B. A portion
enters the cylinder D through a valve chest similar to that in a
steam engine forces out the piston and causes the wheel, W,
to revolve, thereby doing work and cooling the air. The cooled
air escapes from the valve chest and circulates through the lique-
fying chamber L, where it causes the portion of compressed air
entering at B to liquefy. It then issues from the liquefier and
traverses M, cooling the entering portion of air in A, and finally
returns to the compressor. The pressure of the air when it issues
from D is almost atmospheric, and its temperature is below
— 140° C. About twenty-five per cent of the power consumed

* Compt. rend., II, 500 (1900); I, 1568 (1902); II, 762, 823 (1905).

116

THEORETICAL CHEMISTRY

in compression is regained by the motor. The apparatus pro-
duces about 1 liter of liquid air per horse-power hour. By means
of this improved apparatus, based upon the regenerative principle,
all known gases have now been liquefied, the last to succumb being

Liquid Air
40atmoB..-140°

Fig. 26.

helium which was liquefied in 1908, by Kammerlingh Onnes in the
Ley den cryogenic laboratory. The subjoined table gives the
critical data together with the boiling and freezing temperatures
of some of the more common gases.

CRITICAL DATA FOR GASES.

Gas.

Crit. Temp.

Crit. Press.

(Atmos.)

Boiling Point
(at 760 mm )

Freezing
Point.

Helium

-267°

Hydrogen

-242°

-252° 5

-258° 9

Air

-140°

-191°

Nitrogen

-146°

-195° 5

-210° 5

Oxygen

-118°

50 8

-182° 8

-227°

Carbon monoxide ....
Nitric oxide
Carbon dioxide . .

-141°
- 96°
-f 31°

36
64
73

-190°
-153° 6

- 78°

-207°
-W7°
- 65°

Hydrochloric acid
Ammonia

+ 51°. 3
+130°

81 5
115

- 35°
- 33°. 7

-116°

- 77°

LIQUIDS 117

Vapor Pressure of Liquids. According to the kinetic theory
there is a continuous flight of particles of vapor from the surface
of a liquid into the free space above it. At the same time the
reverse process of condensation of vapor particles at the surface
of the liquid is taking place. Eventually a condition of equilib-
rium will be established between the liquid and its vapor, when
the rate of escape will be exactly counterbalanced by the rate of
condensation of vapor particles. The pressure exerted by the
vapor of a liquid when equilibrium has been attained is known as
its vapor pressure. The equilibrium between a liquid and its vapor
is dependent upon the temperature. For every temperature
below the critical temperature, there is a certain pressure at which
vapor and liquid may exist in equilibrium in all proportions; and
conversely for every pressure below the critical pressure, there is
a certainjbemperature at which vapor and liquid may* exist in
equiirBrium in all proportions. This latter temperature is termed
the Boiling-point of the liquid. The vapor pressure of a liquid
may be measured directly by placing a portion of it above the
mercury in the vacuum of a barometer tube, heating to the desired
temperature, and observing the depression of the mercury column.
This is known as the static method. It is open to the objection
that the presence of volatile impurities in the liquid causes too
great depression of the mercury column, the vapor pressure of
the impurity adding itself to that of the liquid whose vapor
pressure is sought. A better method for the measurement of
vapor pressure is that known as the dynamic method. In this
method the pressure is maintained constant and the boiling
temperature is determined with an accurate thermometer. The
boiling temperatures corresponding to various pressures may be
measured, provided we have a suitable device for changing and
measuring the pressure. The results obtained by the static and
dynamic methods agree closely if the liquid is pure, but if volatile
impurities are present the results obtained by the dynamic method
are more trustworthy. A method for the measurement of vapor
pressure due to James Walker * is of considerable interest. In
this method, a current of pure dry air is bubbled through a weighed

* Zeit. phys. Chem., 2, 602 (1888).

118 THEORETICAL CHEMISTRY

amount of the liquid whose vapor pressure is to be determined.
The liquid is maintained at constant temperature and its loss in
weight is observed. In passing through the liquid the air will
absorb an amount of vapor directly proportional to the vapor
pressure of the liquid. If 1 mol. of liquid is absorbed by v liters
of air, then we have

pv = RT,

where p is the vapor pressure of the liquid, and T its temperature.
If Vi is the volume of air which absorbs g grams of the vapor of
the liquid whose molecular weight is M , then

In this equation, Vi denotes the total volume containing g grams
of the liquid in the form of vapor, or in other words it represents
the air and vapor together. Since the volume of the air is in
general so much greater than that of the vapor, v\ may be taken
as that of the air alone.

Heat of Vaporization. In order to transform a liquid into a
vapor a large amount of heat is required. Thus, when a liquid
is heated to the boiling-point, the volume must be increased against
the pressure of the atmosphere, external work being done, and
when the boiling temperature is reached the liquid must be vapor-
ized; the heat expended in causing the change of physical state
being much greater than that required to expand the liquid. An
interesting relation between the heat of vaporization and the
absolute boiling-point of a liquid was discovered by Trouton.*
If T denotes the absolute boiling-point and w the heat of vapor-
ization of 1 gram of liquid whose molecular weight is M, then
according to Trouton

Mw _

-y~ = ^1,

or in words, the ratio of the molecular heat of vaporization to the
absolute boiling temperature of a liquid is constant, the numerical

* Phil. Mag. (5), 18, 54 (1884).

LIQUIDS

119

value of the ratio being approximately 21. This is known as
Trouton's law. While this relation holds quite well for many
liquids, Nernst has pointed out that the constant varies with the
temperature, and has proposed two other forms of the equation.
Bingham has simplified the [equations of Nernst to the following
form: —

^ = 17 + 0.011 T.

While this modification of the Trouton equation has been found
to hold for a large number of substances, there are other substances
for which the left side of the equation has a value greater than
that of the right side. Bingham infers that where this occurs, the
substance in the liquid state has a greater molecular weight
than it has in the gaseous state, or in other words, the liquid is
associated. It is evident that an associated liquid will require
an expenditure of energy over and above that required for vapori-
zation, to break down the molecular complex. The difference
between the values of the two sides of the equation may be
taken as a rough measure of the degree of association.

Boiling-Point and Critical Temperature. An interesting rela-
tion has been pointed out by Guldberg * and Guye.f These
two investigators have shown that the absolute boiling temper-
ature of a liquid is about two-thirds of its critical temperature.
That this empirical relation holds for a variety of different sub-
stances is shown in the accompanying table.

RELATION OF BOILING-POINT TO CRITICAL TEMPERATURE.

Substance.

JVTV,

Oxygen

90°

155°

0.58

Chlorine

240°

414°

0 58

Sulphur dioxide

263°

429°

0.61

Ethyl ether

308°

467°

0.66

Ethyl alcohol . .

351°

516°

0.68

Benzene

353°

562°

0 63

Water

373°

637°

0.59

Phenol .

454°

691° '

0.66

* Zeit. phys. chem., 5, 376 (1890).
f Bull. Soc. Chim., (3), 4, 262 (1890).

120 THEORETICAL CHEMISTRY

Molecular Volume. In dealing with the volume relations of
liquids it is customary to employ the molecular volume, i.e., the
volume occupied by the molecular weight of the liquid in grams.
The justification of this procedure is that when we compare the
gram-molecular weights of liquids, the comparison involves equal
aumbers of molecules of the different substances. Since

, mass

volume =

density '
we may write

, molecular weight

molecular volume = — —. — r- — a— >

density

and similarly,

atomic weight

atomic volume =

density

Relations between the molecular volumes of liquids were first
pointed out by Kopp.* On comparing the molecular volumes of
different liquids at their boiling-points, he found that constant
differences in composition correspond to constant differences in
the molecular volumes. Thus the molecular volumes of the
successive members of an homologous series differ by the same
number of units, this difference corresponding, for example, to a
CH2 group. In like manner the molecular volumes of various
groups have been determined, and from these in turn the atomic
volumes of the constituent elements have been worked out. The
atomic volumes assigned by Kopp to some of the elements com-
monly entering into organic compounds are as follows : —

C = 11 Cl = 22 8 1=375 Hydroxyl 0 = 7.8

H = 5 .5 Br = 27 .8 S = 22 6 Carhonyl 0 = 12 .2

The value of the atomic volume is found to be dependent upon
the manner of linkage; thus oxygen in the hydroxyl group has the
atomic volume, 7.8, while oxygen in the doubly linked condi-
tion, as in the carbonyl group, has the atomic volume, 12.2. By
means of such a table of experimentally determined atomic

* Lieb. Ann., 41, 79 (1842); 96, 153, 303 (1855); 96, 171 (1855).

LIQUIDS 121

volumes, Kopp showed that it is possible to calculate the molec-
ular volume of a liquid with a fair degree of accuracy. For
example, the molecular volume of acetic acid C2H402, may be
calculated from the atomic volumes of its constitutent atoms as
follows: —

2 C = 2 X 11 =22
4H = 4 X 5.5 = 22
IHydroxylO = 1 X 7.8 = 7.8
1 Carbonyl 0 = 1 X 12 2 = 12.2
Molecular volume = 64 0

The density of acetic acid at its boiling-point is 0.942, and its
molecular weight is 00, therefore the observed value of the molec-
ular volume is 60 -f- 0.942 = 63.7, a result which is in excellent
agreement with that calculated from the atomic volumes of the
constituents. The more recent investigations of Thorpe, Lossen,
Schiff and Buff afford a confirmation of the conclusion reached by
Kopp, that the molecular volumes of liquids are in general additive.
While Kopp found that his results were most regular when the
molecular volumes were determined at the boiling temperatures
of the respective liquids, the reason for this did not appear until
after van der Waals had developed his theory of corresponding
states. As has been pointed out in the preceding paragraph the
boiling-points of most liquids are approximately two-thirds of
their respective critical temperatures, and therefore the boiling-
points are corresponding temperatures.

Co-volume. By studying various series of hydrocarbons,
alcohols and ethers, Traube * has been led to suggest that the
molecular volume of a liquid be looked upon as made up of the
atomic volumes of its constituent elements and a magnitude
which he terms the co-volume. This latter he defines as the space
surrounding a molecule within which it is free to vibrate and from
which other molecules are excluded. The co- volume appears to
be nearly constant for a large number of substances, its mean
value at a temperature of 15° C. being 25.9 cc. The values

* Uber den Raum der Atome. J. Traube. Ahrens' Sammlung Chemischer
und chemisch-technischer Vortraege, 4, 255 (1899).

122 THEORETICAL CHEMISTRY

assigned by Traube to the atomic volumes of some of the elements
are as follows: —

C = 9 .9 0=5.5 Br = 17.7 N (trivalent) = 1 .5
H = 3 . 1 Cl = 13 .2 I = 21 A N (pentavalent) = 10 .7
Traube has worked out a series of constants which must be de-
ducted to allow for ring formation and for double and treble link-
ing. By means of these values, it is possible to calculate the
molecular volume of a substance by adding together the respective
atomic volumes of the constituents of the liquid and the co-
volume, 25.9. It is of course necessary to know the molecular
weight of the substance together with its constitution, so that
due allowance may be made for unsaturation. For example,
the molecular volume of ethyl ether, C4HioO, may be calculated
by Traube's method as follows: —

4C = 4X9 9 = 39.6
10H = 10X3.1 = 31
10 = 1 X 5.5 = 5.5
76.1

Co- volume 25 . 9
Molecular volume 102.0

The molecular volume, as determined from the molecular weight
and density at 15° C., is 74 -f- 0.7201 = 102.7.

The method of Traube may be employed in roughly checking
the accepted value of the molecular weight of a liquid provided
its density at 15° C. is known, since in the equation

M/d = £ atomic volumes + 25.9, expressing Traube's relation,

M is the only unknown quantity. It is apparent that the liquid
must be non-associated, since for an associated substance the
normal co-volume must necessarily accompany the polymerized
molecule. In this case the formula becomes

M/d = S atomic volumes + 25.9/n,

where n denotes the number of simple molecules in the polymer.
Obviously when the molecular weight of a liquid is known, the
experimental determination of the co-volume, (M/d — S atomic
vols.) may be used to estimate the degree of association. The

LIQUIDS

values thus obtained are not in satisfactory agreement with the
factors of association derived by means of other methods.

Refractive Power of Liquids. The velocity of transmission
of light through any medium depends upon its nature, especially

Fig. 27.

upon its density. When a ray of light passes from one medium
into another it is refracted, the degree of refraction being such
that the ratio of the sines of the angles of incidence and refrac-
tion is constant and characteristic for the two media. This

124

THEORETICAL CHEMISTRY

fundamental law of refraction was discovered by Snell about
1621. According to the wave theory of light, the ratio of the
sines of the angles of incidence and refraction is identical with
the ratio of the velocities of light in the two media. The ratio is
termed the index of refraction and is usually denoted by the letter
n. Representing by i and r, the angles of incidence and refraction,
and by v\ and v%, the respective velocities of light in the two media,

we have

sin i V]

n = -T— = —
sin r v%

Various forms of apparatus have been devised for the determina-
tion of the refractive index of liquids. Of these the best known
and most satisfactory is the refractometer of Pulfrich, an improved
form of which is shown in Fig. 27. While the limits of this book
prohibit a detailed description of the apparatus, the fundamental
principles involved in its construction will be readily understood
from the accompanying diagram, Fig. 28. The liquid or fused
solid is placed in a small glass cell, C, which is cemented to a rec-
tangular prism of dense optical
glass, P, the refractive index of
which is generally 1.61. A beam
of monochromatic light, from a
sodium flame of a spectrum-tube
containing hydrogen, is allowed to
enter the prism in a direction par-
allel to the horizontal surface of
separation between the glass and
the liquid. After passing through

Fig. 28.

the liquid and the prism, the beam emerges making an angle i with
its original direction. By means of a telescope, the emergent beam
can be observed and its position noted, the angle of emergence
being read on a divided circle attached to the telescope. From the
angle of emergence thus determined, the index of refraction of the
liquid can be calculated in the following manner. The value of
the index of refraction, N, for air/glass being known, we have

sin i
sin r

(i)

LIQUIDS 125

The angle of incidence of the last ray entering the prism from the
liquid is 90°, or sin ii = 1. The index of refraction, rai, for liquid/
glass may be calculated thus,

sin

n —

But __

(3)

Transposing equation (1) and substituting in equation (3), we
have

sin2i
smn =

_ _ __ __ .

sin 7*1 = -T vW2 — sin2i. (4)

Therefore, substituting equation (4) in equation (2), we have

Remembering that n = JV/ni, we have for the index of refraction,
n, for air/liquid, by substitution in equation (5),

n =
or if N = 1.61, ______

w = V2.592i-sin2i'.

The values of ^/N* — sin2! are generally given, for different values
of i, in tables supplied with the refractometer, thus saving the
experimenter a somewhat laborious calculation. The value of n
thus obtained is the index of refraction from air into the liquid; if
the index from vacuum into the liquid, the so-called absolute index,
is required, the value of n must be multiplied by 1.00029. The
index of refraction is dependent upon temperature, pressure and
in general upon all conditions which affect the density of the
medium. Furthermore, it is dependent upon the wave-length of
the light employed, the index for the red rays being greater than
that for the violet rays. It is therefore necessary in making

126 THEORETICAL CHEMISTRY

measurements of refractive indices to use light of a definite wave-
length, or what is termed monochromatic light. The sodium
flame is most frequently used for this purpose, the wave-length
being represented by the letter D. Measurements of the refrac-
tive index referred to the D-line of sodium are commonly desig-
nated by the symbol n/>. When incandescent hydrogen is employed
as a source of light, the refractive index may be determined
for the C-, F- and G-lines, the respective values being represented
by nc, nF, and UQ.

Specific and Molecular Refraction. Various attempts have
been made to express the refractive power of a liquid by a formula
which is independent of variations of temperature and pressure.
Of the different formulas proposed but two need be mentioned.
The first, due to Gladstone and Dale,* is as follows: —

d '

in which d denotes the density of the liquid and n is the so-called
specific refraction. The other formula, proposed by Lorenz f and
Lorentz,{ has the following form: —

__ 1 n2- 1
r2 ~d'n* + 2'

This formula is superior to that of Gladstone and Dale which is
purely empirical. It is based upon the electromagnetic theory
of light and gives values of r2 which are quite independent of the
temperature. In order that we may compare the refractive
powers of different liquids, the specific refractions are multiplied
by their respective molecular weights, the resulting products
being termed their molecular refractions. As the result of a large
number of experiments, it has been shown that the molecular
refraction of a compound is made up of the sum of the refractive
constants of the constituent atoms, or in other words refractive
power is an additive property. The values of the refractive con-
stants of the elements and commonly-occurring groups have been

* Phil. Trans. (1858).

t Wied. Ann., n, 70 (1880).

t Ibid., 9, 641 (1880).

LIQUIDS 127

determined with great care by Brtihl and others, the method
employed being similar to that used by Kopp in connection with
his investigations on molecular volumes. Thus, Brtihl found in
the homologous series of aliphatic compounds that a difference
of CH2 in composition corresponds to a constant difference of
4.57 in molecular refraction. Then, having determined the
molecular refraction of a ketone or an aldehyde of the composition,
CnH2»0, he subtracted n times the value of CH2 and obtained the
atomic refraction of carbonyl oxygen. By deducting the molecular
refraction of the hydrocarbon, Cn#2n+2, from that of the corre-
sponding alcohol, Cn//2n+20, he obtained the atomic refraction of
hydroxyl oxygen. By subtracting six times the value of CH2
from the molecular refraction of hexane, C6Hi4, he obtained the
refractive constant for hydrogen or 2 H = 2.08. In like manner
the refractions of other elements and groups of elements were
determined.

Just as in the case of molecular volumes so with molecular
refractions, the arrangement of the atoms in the molecule must
be taken into consideration. Briihl,* who has devoted much time
to the investigation of the effect of constitution upon refraction,
has pointed out that the molecular refraction of compounds conT
taining double and triple bonds is greater than the calculated value,
and he has assigned to these bonds definite constants of refrac-
tion. The values of the atomic refractions for a few of the elements
as given by Brtihl are as follows: —

C = 2.48 Hydroxyl O = 1.58

H= 1.04 Carbonyl O -2.34

CI = 6 .02 Double bond = 1 .78

I = 13 . 99 Triple bond =2.18

More recent investigations bring out the fact that, when double
or triple bonds occupy adjacent positions in the molecule, the
simple additive relations no longer obtain. The determination
of the molecular refraction of a liquid affords a means of ascertain-
ing or confirming its chemical constitution. For example, geraniol
has the formula CioHisO, and its chemical behavior is such as to

* Proc. Roy. Inst., 18, 122 (1906).

128 THEORETICAL CHEMISTRY

warrant the conclusion that it is a primary alcohol. The value
of UD is 1.4745, from which the molecular refraction is calcu-
lated to be 48.71. The molecular refraction calculated from the
atomic refractions given in the preceding table is: —

IOC = 10 X 2.48 = 24.80

18 H = 18 X 1.04 = 18 72

1 Hydroxyl O = 1 X 1 58 = 1_58

Molecular refraction 45 . 10

The difference between the theoretical and experimental values
of the molecular refraction is 48.71 — 45.10 = 3.61, which is
approximately twice the value of a double bond, 1.78 X 2 = 3.56.
From this we conclude that the molecule of geraniol contains two
double bonds. Furthermore an alcohol of the formula, ( a0Hi8(),
containing two double bonds cannot possess a ring structure and
therefore must be a member of the aliphatic group of compounds.
This conclusion is supported by the chemical properties of the
substance.* In a similar manner the Kekule formula for benzene
has been confirmed, the difference between the theoretical and
experimental -values of the molecular refraction indicating the
presence of three double bonds in the molecule.

Specific Refraction of Mixtures. The specific refraction of an
homogeneous mixture or solution is the mean of the specific
refractions of its constituents. Thus, if the specific refractions
of the mixture and its two components are represented by r\, r2,
and r3, then

- P . 000 -p)

~ ""~

where p denotes the percentage of the constituent whose specific
refraction is r2. Hence it is possible to determine the specifiq
refraction of a substance in solution by measuring the refractive
indices and densities of the solution and solvent. If the refrac-
tive indices of the solvent, solution and dissolved substance are

* The accepted structural formula of geraniol is

H - C - CH2OH

II
(CH3)2C - CH-CH2-CH2 - C - CH8.

LIQUIDS 129

represented by n\, ^ and n3 respectively, and if di, da, and da
denote the corresponding densities, then we have

JZJ: _ 1QQ"" P V-- 1
"

where p is the percentage of the dissolved substance. As has
already been mentioned, the formula of Lorenz-Lorentz is based
upon the electromagnetic theory of light. According to this
theory n* — 1/n2 + 2 expresses the fraction of the unit of volume
of the substance which is actually occupied by it. From this it

follows that the molecular refraction, — • -, is an expression

d ti -]- &

of the volume actually occupied by the atomic nuclei of the
molecule. It is interesting to note that the ratio of the sum of
the atomic volumes, calculated by the method of Traube, to the
corrected molecular volume, as determined by the Lorenz-Lorentz
formula, is approximately constant, or

S atomic volumes 0 . r . , ,

- rr — run - ^ 3-45 approximately.

This may be considered as the ratio of the volume within which
the atoms execute their vibrations to their actual material volume.
Rotation of the Plane of Polarized Light. Some liquids when
placed in the path of a beam of polarized light possess the prop-
erty of rotating the plane of polarization to the right or to the
left. Such liquids are said to be optically active. Those substances
which rotate the plane of polarization to the right are termed
dextro-rotatory, while those which cause an opposite rotation are
called levo-rotatory. The determination of the rotatory power of
a liquid is made by means of an instrument known as a polarimeter,
a convenient form of which is shown in Fig. 29. The essential
parts of this instrument are two similar Nicol prisms placed one
behind the other with their axes in the same straight line. The
light after passing through the forward prism, P, known as the
polarizer, has its vibrations reduced to a single plane; it is said
to be plane polarized. On entering the rear Nicol prism, A,

130

THEORETICAL CHEMISTRY

known as the analyzer, the light will either pass through or be
completely stopped, depending upon the position of the prism.
If the analyzer be slowly rotated, it will be observed that
the position of maximum transmission and extinction occur at
points 90° apart. If the analyzer be rotated, so that its axis is
at right angles to the axis of the polarizer, the field observed will
be dark, no light being transmitted. If now a tube similar to
that shown in Fig. 30 be filled with an optically active liquid
and placed between the polarizer and analyzer, the field will

Fig. 29.

become light again, due to the rotation of the plane of polariza-y
tion by the optically-active substance. The extent to which the
plane of polarization has been rotated can be determined by
turning the analyzer until the field becomes dark again, and read-
ing on the divided circle, K , the number of degrees through which
it has been moved. When it is necessary to turn the analyzer
to the right, the substance is dextro-rotatory, and when it is neces-
sary to turn it to the left, the substance is levo-rotatory. Various

LIQUIDS 131

optical accessories have been added to the simple polarimeter
described above to render the instrument more sensitive, but for
these details the student must consult some special treatise.*
The angle of rotation is dependent upon the nature of the liquid,
the length of the column of substance through which the light
passes, the wave-length of the light used, and the temperature at
which the measurement is made. It is customary in polarimetric

Fig. 30.

work to employ sodium light and, unless otherwise specified, it
may be assumed that a given rotation corresponds to the D-line.

Specific and Molecular Rotation. The results of polarimetric
measurements are expressed either as specific rotations or as
molecular rotations, the latter being preferable since the optical
activities of different substances may then be compared.

The specific rotation is obtained by dividing the observed rota-
tion by the product of the length of the column of liquid and its
density, or

where [a]t is the specific rotation at the temperature, t, a the
observed angle, I the length of the column of liquid in decimeters,
and d its density. If the specific rotation is multiplied by the
molecular weight of the substance, the molecular rotation is ob-
tained, but owing to the fact that the resulting numbers are too
large, it is customary to express the molecular rotation as one
one-hundredth of this value, thus

100H

The specific and molecular rotations of solutions of optically
active substances may also be determined, if we assume that the

* See for example, 4CThc Optical Rotatory Power of Organic Substances
and its Practical Applications." H. Landolt, trans, by J. H. Long.

132

THEORETICAL CHEMISTRY

solvent is without effect. While this assumption is justifiable
with aqueous solutions, it is not so when non-aqueous solvents are
used. If g grams of an optically active substance be dissolved in
v cc. of solvent, then

r , av j r i M av
[«],-•£, a*d [«„],= _._,

or if the composition of the solution is expressed in terms of
weight instead of volume, g grams of substance being dissolved
in 100 grams of solution of density d, then

100 a.

gdl '

and

Optical Activity and Chemical Constitution. The fact that
some substances have the power of rotating the plane of polarized
light was first discovered by Biot, but the credit for recognizing
the chemical significance of this fact belongs to Pasteur.* He
discovered that ordinary racemic acid can be separated into two
optically active modifications, one of which is dextro- and the
other levo-rotatory, the numerical values of the two rotations

Fig. 31.

being identical. If a solution of sodium ammonium racemate
be allowed to evaporate at a low temperature, crystals of the
composition NaNH^EUOe . 4 H2O will separate. On close in-
spection it will be found that the crystals are not all alike, but
that they may be divided into two classes, one class showing
some unsymmetrical crystal surfaces which are oppositely placed
in the crystals of the other class. The crystals of one class may

* Ann. Chim. Phys. (3), 24, 442 (1848); 28, 56 (1850); 31, 67 (1851).

LIQUIDS 133

be regarded as the mirror images of those of the other class:
such crystals are said to be enantiomorphous. The forms usually
assumed by the two enantiomorphous modifications of sodium
ammonium racemate are shown in Fig. 31. After separating
the two forms Pasteur dissolved each in water, making the solu-
tions of the same strength. The solution of the crystals with the
"right-handed faces" was found to be dextro-rotatory, while that
of the crystals with the " left -handed faces" was found to be levo-
rotatory. Pasteur then decomposed the two salts obtained from
sodium ammonium racemate and obtained the corresponding
acids, which he called dextro- and levo-racemic acids. It was
subsequently shown that the two acids were identical with dextro-
and levo-tartaric acids. Finally, when Pasteur mixed equiv-
alent amounts of concentrated solutions of dextro- and levo-
tartaric acids, an appreciable evolution of heat was observed,
indicating that a chemical reaction had taken place. After
allowing the solution to stand for some time, crystals of ordinary
racemic acid were obtained. Thus it was clearly proven that an
optically inactive substance may be separated into two opti-
cally active modifications, possessing equal and opposite rotatory
powers, and that by mixing equivalent quantities of the two
optically active forms, the optically inactive substance may be
recovered.

Pasteur discovered and applied three other methods in addi-
tion to the mechanical method already described, for the separation
of a substance into its optically active modifications. These are
as follows: — (a) Method of Crystallization; (b) Method of Forma-
tion of Derivatives; and (c) Methods of Ferments.

Methods of Crystallization. To a supersaturated solution of the
racemic modification a very small crystal of one of the active
forms is added. This will induce the separation of crystals of
the same form, inoculation with a dextro-crystal producing the
dextro-form and inoculation with a levo-crystal producing the
levo-form.

Method of Formation of Derivatives. In this method an opti-
cally active substance, generally an alkaloid, is added to the racemic
modification, producing optically active derivatives having differ-

134 THEORETICAL CHEMISTRY

ent solubilities. Thus if cincbonine, an optically active alkaloid
having the formula, CigEM^O, be added to the racemic modifica-
tion of tartaric acid, the cinchonine salt of the levo-acid will
crystallize first. The crystals of the cinchonine salt are then
removed and after adding ammonia to displace the alkaloid,
dilute sulphuric acid is added and the pure levo-tartaric acid is
obtained.

Methods of Ferments. Notwithstanding the fact that optical
antipodes resemble each other so closely in most of their properties,
Pasteur found that certain micro-organisms have the power of
distinguishing sharply between these forms. For example, if
penicillium glaucum be introduced into a solution of racemic
tartaric acid, it thrives at the expense of the dextro-acid and
eventually leaves the pure levo-form. In this method one of the
active modifications is always lost.

Pasteur was the first to point out that there must be some inti-
mate connection between optical activity and the constitution of
the molecule. It remained for Le Bel * and van't Hoff f to for-
mulate independently and almost simultaneously an hypothesis
to account for optical activity on the basis of molecular constitu-
tion. Their important work laid the foundation of spatial chemis-
try, commonly termed stereochemistry (derived from the Greek
orrcpeos = a solid). Le Bel accepted Pasteur 's view that optical
activity is dependent upon a condition of asymmetry, but whether
'this asymmetry is a property of the crystal alone or whether it
belongs to the molecule of the optically active substance, was the
question he set himself to answer. He found, on dissolving certain
optically active crystals in an inactive solvent, that the optical
activity is imparted to the solution and therefore he concluded
that the condition of asymmetry must exist in the chemical mole-
cule. All of the optically active substances known to Le Bel were
compounds of carbon. An examination of the formulas of these
compounds led him to ascribe the cause of their optical activity
to the presence of an asymmetric carbon atom, that is, a carbon
atom combined with four different atoms or groups of atoms.

* Bull. Soc. Chim, (2), 22, 337 (1874).
t Ibid. (2), 23, 295 (1875).

LIQUIDS

135

One of the simplest examples is afforded by lactic acid, the struc-
tural formula of which is

CH3 — C — COOH

In this formula the asymmetric carbon atom is placed at the
center and is in combination with hydrogen, hydroxyl, methyl
and carboxyl. In connection with his work on the relation
between optical activity and asymmetry, Le Bel pointed out that
active forms never result from laboratory syntheses, the racemic
modification being invariably obtained. Van't Hoff reached con-
clusions similar to those of Le Bel and proposed the additional

COOH

Fig. 32.

theory of the asymmetric tetrahedral carbon atom. Since the four
valences of the carbon atom are equivalent, as the work of Henry
on methane has shown them to be, van't Hoff pointed out that
the only possible geometrical arrangement of the atoms in the
molecule of methane must be that in which the carbon atom is
placed at the center of a regular tetrahedron with the four
hydrogen atoms at the four apices. He then pointed out that
when the four valences of the tetrahedral carbon atom are satis-
fied with different atoms or groups, no plane of symmetry can be
passed through the figure, the carbon atom being asymmetric.
This conception of Le Bel and van't Hoff forms the basis of all
stereochemistry, and has proved of inestimable value to the
organic chemist in enabling him to explain the existence of many
isomeric compounds. Thus, ordinary lactic acid can be split into

136

THEORETICAL CHEMISTRY

two optically active isomers. Aside from the fact that one acid
is dextro- and the other is levo-rotatory, the properties of the
two acids are practically identical. If the formulas are written
spatially, the different groups can be arranged about the
asymmetric carbon atom in such a way that the two tetrahedra
shall be mirror images of each other, as shown in Fig. 32. It will
be observed that these two tetrahedra can in no way be super-
posed so that the same groups fall over each other, that is to
say, they are enantiomorphous forms. In tartaric acid there are
two asymmetric carbon atoms as is evident when its structural
formula is written as follows: —

H H

HOOC — C — C — COOH
OH OH

If the stereochemical formulas of the dextro- and levo-acids be
represented as in Fig. 33, (a) and (b), it will be apparent that
the theory admits of the existence of another isomer with the
atoms and groups arranged as in Fig. 33 (c).

Bacemic Acid
A

GO OH
d-TartaricAcid

«*>

COOH
2-Tartaric Acid

ib)
Fig. 33.

COOH
Meso-Tartaric Acid

In this arrangement the asymmetry of the upper tetrahedron
is the reverse of that of the lower, and consequently the optical
activity of one-half of the molecule exactly compensates the optical

LIQUIDS 137

activity of the other half, and the molecule as a whole is inactive.
It is evident that such a tartaric acid could not be split into two
active forms. Actually there are four tartaric acids known, viz.,
(1) inactive racemic acid which is separable into (2) dextro-tartaric
acid and (3) levo-tartaric acid; and (4) meso-tartaric acid, an in-
active substance which has never been separated into two active
forms, but which has the same formula, the same molecular
weight and in general the same properties as the dextro- or levo-
tartaric acids. Inactive forms, such as meso-tartaric acid, are said
to be inactive by internal compensation. This constitutes one of
many beautiful confirmations of the van't Hoff theory of the
asymmetric tetrahedral carbon atom.

Meso-tartaric acid furnishes an illustration of the fact that
asymmetric carbon atoms may be present in the molecule with-
out imparting optical activity to the substance. The converse
of this proposition, however, that optical activity is dependent
upon asymmetric carbon atoms, is generally true. Quite recently
some substances apparently containing no asymmetric carbon
atoms have been discovered which are optically active. An
example of such a substance is 1-methyl cyclohexylidene-4 acetic
acid, to which the following formula has been assigned: —

CH2 • CH2

CH3CH C : CH • COOH

CH2 • CH2

Other atoms aside from carbon may be asymmetric; thus certain
compounds of nitrogen, sulphur and tin have been shown to be
optically active. The theory also furnishes an explanation of
the fact, pointed out by Le Bel, that optically active forms are
never obtained by direct synthesis. Since the rotatory power is
dependent upon the arrangement of the atoms and groups in the
molecule, it follows from the doctrine of probability that as many
dextro as levo configurations will be formed and consequently the
racemic modification will be obtained. Up to the present time no
satisfactory generalization has been discovered as to the factors
determining the molecular rotation in any particular case. An

138

THEORETICAL CHEMISTRY

attempt in this direction has been made by Guye,* in which he
ascribes the magnitude of the observed rotation to the relative
masses of the atoms or groups which are in combination with the
tetrahedral carbon atom. But it cannot be mass alone which
conditions optical activity, since substances are known which
rotate the plane of polarization notwithstanding the fact that
their molecules have two groups of equal mass in combination
with the asymmetric carbon atom. The molecular rotations of
the members of homologous series exhibit some regularities, but
on the other hand many exceptions occur which cannot be satis-
factorily explained. About all that can be said at the present
time is, that optical activity is a constitutive property.

Magnetic Rotation. That many substances acquire the power
of rotating the plane of polarized light when placed in an intense
magnetic field was first observed by Faraday f in 1846.

The relation between chemical composition and magnetic
rotatory power has since been investigated very exhaustively by
W. H. Perkins,! his experiments in this field having been continued
for more than fifteen years. In brief, Perkin's method of investi-
gating magnetic rotatory power consisted in introducing the
liquid to be examined into a polarimeter tube 1 decimeter in
length and then placing the tube axially
between the perforated poles of a
powerful electromagnet, as shown in
Fig. 34. Upon exciting the magnet it
was found that the plane of polarization
had been rotated, either to the right
or the left, the direction of rotation
depending upon the direction of the
current, the intensity of the magnetic

Fig. 34.

field and the nature of the liquid. Perkin used the sodium flame
as his source of light and carried out all of his experiments
at 15° C. He expressed his results by jneans of the formula,

* Compt. rend., no, 714 (1890).

t Phil. Trans., 136, 1 (1846).

j Jour, prakt. Chem. [2], 31, 481 (1885); Jour. Chem. Soc., 49, 777; 41,
808; 53, 561, 695; 59, 981; 6x, 287, 800; 63, 57; 65, 402, 815; 67, 255; 69,
1025 (1886-1896).

LIQUIDS 139

Ma/dj a being the observed angle of rotation, d the density
of the liquid and M its molecular weight. All measurements
were expressed in terms of water as a standard: thus if Ma/d
is the rotation for any substance and M'a! '/d' is the corre-
sponding rotation for water, then, according to Perkin, the
molecular magnetic rotation will be given by the ratio, Ma/d:
M'a'/d' or Mad'/M'a'd.

The molecular magnetic rotation for a large number of organic
compounds has been determined by Perkin, who has shown it to
be an additive property. In any one homologous series the value
of the molecular magnetic rotation is given by the formula

mol. mag. rotation = a + rib,

where a is a constant characteristic of the series, b is a constant
corresponding to a difference of CH2 in composition, its value
being 1.023, and n is the number of carbon atoms contained in
the molecule. This formula is applicable only to compounds
which are strictly homologous, isomeric substances in two differ-
ent series having quite different rotations. The constitution of
the molecule exerts as great an influence on magnetic rotation
as it does on refraction, a double bond causing an appreciable
increase in the value of a. The results of experiments on mag-
netic rotation show that nothing like the same regularities exist
as have been discovered for molecular refraction and molecular
volume. The rotatory powers of various inorganic substances
have been determined, but the results are too irregular to admit
of any satisfactory interpretation.

Absorption Spectra. When a beam of white light is passed
through a colored liquid or solution and the emergent beam is
examined with a spectroscope, a continuous spectrum crossed by
a number of dark bands is obtained. A portion of the light has
been absorbed by the liquid. Such a spectrum is known as an
absorption spectrum. If instead of passing the light through a
liquid it is passed through an incandescent gas, a spectrum will
be obtained which is crossed by numerous fine lines, termed
Fraunhofer lines. Such lines occupy the same positions as the
corresponding colored lines in the emission spectrum of the gas.

140

THEORETICAL CHEMISTRY

It follows, therefore, that the absorption spectrum is quite as char-
acteristic of a substance as its emission spectrum, and from a
careful study of the absorption spectra of liquids we may expect
to gain some insight into their molecular constitution. The
pioneer workers in this field were Hartley and Baly * and it is
largely to them that we owe our present experimental methods.
The instrument employed for photographing spectra is called a

Fig. 35.

spectrograph, a very satisfactory form being shown in Fig. 35.
It differs from an ordinary spectroscope in that the eye-piece is
replaced by a photographic camera. This attachment is clearly
shown in the illustration. The plateholder is so constructed that
only a narrow horizontal strip of the plate is exposed at any one
time, thus making it possible to take a series of photographs on
the same plate by simply lowering the holder. By means of a
millimeter scale, also shown in the illustration, the plateholder
can be moved through the same distance each time before expos-

* See numerous papers in the Jour. Chem. Soc., since 1880.

LIQUIDS 141

ing a fresh portion of the plate, thus insuring an equally-spaced
series of spectrum photographs. In order that spectra in the ultra-
violet region may be photographed, it is customary to equip the in-
strument with quartz lenses and a quartz prism, ordinary glass not
being transparent to the ultra-violet rays. Using a spectrograph
furnished with a quartz optical system, it is possible to photograph
on a single plate the entire spectrum from 2000 to 8000 Angstrom
units. A scale of wave-lengths photographed on glass is provided
with the instrument so that the wave-lengths of lines or bands
can be read off directly by laying the scale over the photographs.

The source of light to be used depends upon the character of
the investigation. If a source rich in ultra-violet rays is desired,
the light from the electric spark obtained between electrodes pre-
pared from an alloy of cadmium, lead and tin is very satisfactory;
or the light from an arc burning between iron electrodes may be
used. For investigations in the visible region of the spectrum
the Nernst lamp is unsurpassed. In using the spectrograph for
the purpose of studying the constitution of a dissolved substance,
it is necessary to determine not only the number and position of
the absorption bands, but also the persistence of these bands as
the solution is diluted.

According to Beer's law the product of the thickness, t, of an
absorbing layer of solution of molecular concentration, m, is con-
stant, or mt = fc. If then the thickness of a given layer of solu-
tion is diminished n times, its absorption will be the same as that
of a solution whose concentration is only 1 /nth of that of the original
solution. Thus, by varying the thickness of the absorbing layer
we can produce the same effect as by changing the concentration.
The convenient device of Baly for altering the length of the
absorbing column of liquid is shown in Fig. 36 attached to the
collimator of the spectrograph. It consists of two closely-fitting
tubes, one end of each tube being closed by a plane, quartz disc.
The outer tube is fitted with a small bulbed-funnel and is graduated
in millimeters. The two tubes are joined by means of a piece of
rubber tubing which prevents leakage of the contents, and at the
same time admits of the adjustment of the column of liquid to
the desired length by simply sliding the smaller tube in or out.

142

THEORETICAL CHEMISTRY

Molecular Vibration and Chemical Constitution. There are
two systems of graphic representation of the results of spectro-
soopic investigations. In the first system, due to Hartley, the
wave-lengths or their reciprocals, the frequencies, are plotted as
abscissae and the thicknesses of the absorbing layers, in milli-
meters, are plotted as ordinates. Such curves are known as
curves of molecular vibration. The second system, due to Baly
and Desch, is a modification of that developed by Hartley.

Fig. 36.

Baly and Desch suggested that for various reasons it would be
more advantageous, if instead of plotting the thickness of the
absorbing layers as ordinates, the logarithms of these thicknesses
be plotted. Both methods have their advantages and both are
used. As an illustration of the value of curves of molecular
vibration in connection with questions of chemical constitution,
we will take the case of o-hydroxy-carbanil. The constitution
of this substance was known to be represented by one of the two
following formulas : — '

\NH

C,H«

-OH

(a)

(b)

LIQUIDS

143

On comparing the curves of molecular vibration for the three sub-
stances (Fig. 37), it is apparent that the curves for the lactam
ether and o-hydroxy-carbanil bear a close resemblance to each
other, while the curve for the lactim ether is very different from
the curves for the other two substances. The constitution of

4000

' I I I I I I I I I"

345978912

Oscillation Frequencies

4000 4000

' ' ' I ' ' I I ' ' I I » I « ' I I I I I '
3 4567891

345678y 123

Ether

Q-Hydroxy-CaiibaDil
Fig. 37.

Lactim Ether

o-hydroxy-carbanil must then be very similar to that of the
lactam ether. The formulas of the ethyl derivatives of the
mother substance are known to be as follows: —

- C2H6
Lactam ether

/°\

/ \C

Lactim ether

-0-C2HS

144 THEORETICAL CHEMISTRY

Hartley concluded, therefore, that formula (a), represents the
structure of the molecule of o-hydroxy-carbanil.

It is beyond the scope of this book to discuss at greater length
the bearing of absorption spectra upon chemical constitution; but
the student is earnestly advised to consult some book * treating of
this important subject or to read some of the original papers.

Surface Tension. The attraction between the molecules of a
liquid manifests itself near the surface where the molecules are
subject to an unbalanced internal force. The condition of a
liquid near its surface is roughly depicted in Fig. 38, where the
dots A, J3, and C represent molecules and the circles represent the
spheres within which lie all of the other molecules which exert an
appreciable attraction upon A , B, and C. The shaded portions rep-

Fig. 38.

resent those molecules whose attractions are unbalanced. These
unbalanced forces will evidently tend to diminish the surface to
a minimum value. That is, the contraction of the surface of a
liquid involves the expenditure of energy by the liquid. The
surface film of a liquid is consequently in a state of tension.

Some liquids wet the walls of a glass capillary tube while others do
not. When the liquid wets the tube, the surface is concave and the
liquid rises in the tube; on the other hand, when the liquid does not
adhere, the surface is convex and the liquid is depressed in the tube.
The law governing the elevation or depression of a liquid in a
capillary tube was discovered by Jurin and may be stated thus: —
The elevation or depression of a liquid in a capillary tube is inversely
proportional to the diameter of the tube. Let Fig. 39 represent a

* Relation between Chem. Constitution and Phys. Properties. Samuel
Smiles.

LIQUIDS

145

capillary tube of radius r, immersed in a vessel of liquid whose
density is d, and let the elevation of the liquid in the capil-
lary be denoted by h. Then the weight of the column of liquid
in the capillary will be irrzh dg, where g is the acceleration due
to gravity. The force sustaining this weight is 2 Trry cos 0, the

Fig. 39.

vertical component, of the force due to the tension of the liquid
surface at the walls of the tube, 7 being the surface tension and
0 the angle of contact of the liquid surface with the walls of
the tube.
Therefore

7TT2/l dg = 2 7TT7 COS 0,

_ ^ dgr
7 " 2cos0'

In the case of water and many other liquids 0 is so small that we
may write 0 = 0, the foregoing expression becoming

7 = 1/2 h dgr.

Thus the surface tension of a liquid can be calcuated provided
its density and the height to which it rises in a previously calibrated
tube is known. When h and r are expressed in centimeters, 7
will be expressed in dynes per centimeter or &rgs per square centi-
meter. A simple form of apparatus for the determination of
surface tension used by the author is shown in Fig. 40. A capil-
lary tube, A, of uniform bore is sealed to a glass rod, J5, which is
held in position in the test tube, 5, by means of a cork stopper.
A short right-angled tube, JD, and a thermometer, F, are also

146

THEORETICAL CHEMISTRY

passed through the same cork stopper. The liquid whose surface
tension is to be measured is introduced into the tube, -B, the cork
inserted and the tube placed inside
of the larger tube, C9 containing a
liquid of known boiling-point.
When the thermometer, F, has
become stationary, the capillary
elevation of the liquid is measured
with a cathetometer. The tube,
D, permits the escape of vapor
from the liquid in B and at the
same time insures equality of pres-
sure inside and outside of the ap-
paratus. The spiral tube, <7, serves
as an air condenser, preventing
loss of vapor from the liquid in the
outer tube. The surface tension
of a liquid has been found to depend
upon the nature of the liquid and
also upon its temperature.

Surface Tension and Molecular
Weight. In 1886, Eotvos* showed
that the surface tension multiplied
by the two-thirds power of the mo-
lecular weight and specific volume
is a function of the absolute tem-
perature, or

7 (Mv)* = / (T),

Fig. 40.
where 7 is the surface tension, M

the molecular weight, v the specific volume or reciprocal of the
density, and T the absolute temperature. Ramsay and Shields f
modified the equation of Eotvos as follows: —

7 (MvY =*k(tc-t — 6), (1)

tc being the critical temperature of the liquid, t the temperature
of the experiment, and k a constant independent of the nature of

* Wied. Ann., 27, 448 (1886).

t Zeit. phys. Chem., 12, 431 (1893).

-=3 c

LIQUIDS 147

the liquid. The physical significance of the two-thirds power of
the molecular volume has been explained by Ostwald in the follow-
ing manner: — Assuming the molecules to be spherical, we shall
have for two different liquids, the proportion

Fi:F2::n3:r23,

where V\ and F2 represent the volumes and r\ and r2 the radii of
their respective molecules. Similarly the ratio of the surfaces,
Si and S%, of the molecules in terms of their respective radii, will
be

Si:&::ri«:itf.

From these two proportions it follows that the ratio of the molec-
ular surfaces in terms of the molecular volumes, will be

Si:&::yi*:Ft*.

Making use of the value of M as determined in the gaseous state,
Ramsay and Shields found the value of k for a large number of
liquids to be equal to 2.12 ergs. Among the liquids for which
this value of k was found were benzene, carbon tetrachloride,
carbon disulphide and phosphorus trichloride. For certain other
liquids such as water, methyl and ethyl alcohols and acetic acid,
fc was found to have values much smaller than 2.12. Ramsay
and Shields attributed these abnormalities to an increase in molec-
ular weight due to association, and suggested that the degree of
association might be calculated from the equation

** = 2.12/fc',
or

x = (^rp (2)

where x denotes the factor of association, and fc' is the value of
the constant for the associated liquid in equation (1). It was
further pointed out by Ramsay and Shields that equation (1)
affords a means of calculating the molecular weight of a pure
liquid, provided we assume that for a non-associated liquid the
mean value of fc is 2.12. Since it is not an easy matter to deter-
mine the critical temperature with accuracy, Ramsay and Shields
made use of a differential method, and thus eliminated tc from
equation (1). If the surface tension of a liquid be measured at

148 THEORETICAL CHEMISTRY

two temperatures t\ and fe, and the corresponding densities are d\
and ^2, we shall have

7i (WO* = k (fc-fe-6), (3)

and

72(M/d2)|.==M*c-fe-6). (4)

Subtracting equation (4) from equation (3), we obtain

_ fc = 2.12, (5)

or solving equation (5) for M , we have

The method of Ramsay and Shields is the best known method for
the determination of the molecular weight of a pure liquid. If
M is known to be the same in the liquid and gaseous states,
or in other words, if fc is independent of the temperature, even
though its value is not exactly 2.12, the critical temperature of
the liquid can be calculated by means of equation (1). In order
that the correct value of the critical temperature may be obtained,
Ramsay and Shields found it necessary to use the specific value
of k for the liquid whose critical temperature is sought. As an
illustration of the method of calculation, the following example is
taken from the work of Ramsay and Shields.
For carbon disulphide,

7 at 19°.4 = 33.58 7 at 46°. 1 = 29.41

d at 19°.4 = 1.264 d at 46°.l = 1.223.

We have then for 7 (M/d)l, at the two temperatures,

(76/1.264)* X 33,58 = 515.4,
and

(76/1.223)* X 29.41 = 461.4.

Substituting in the equation

= k'

we have,

515.4-461.4
46.1 - 19.4 ~ 2-022'

LIQUIDS 149

This value of k is so nearly equal to the mean value, 2.12, that we
assume M to be the same in the liquid and gaseous states, and
therefore we may substitute in equation (1) and calculate the
critical temperature of carbon disulphide thus,

or solving for tc, we have

Substituting the data given above, in the preceding equation, we

obtain

tc = 515.4/2.022 + 6 + 19.4,

or te = 280°.3 C.

Surface Tension and Drop-Weight. Morgan and his co-
workers,* from measurements of the volumes of a single drop
falling from the carefully-ground tip of a capillary tube, have
shown that the weight of the falling drop from such a tip can be
used in place of the surface tension in the equation of Ramsay
and Shields for the calculation of molecular weights and critical
temperatures. The modified equation may be written thus: —

where Wi and wz are the respective weights of the falling drop
at the temperatures, t\ and £2- The value of k obviously depends
upon the tip employed.

The results obtained by the drop-weight method have been
shown to be more trustworthy than those obtained by the method
of capillary elevation. Morgan has further pointed out that
when the experimental data are substituted in the preceding
formula, the magnification of the experimental errors is appreci-
ably greater than when use is made of the original formula,

w(M/d)l = fc(«c-«-6).

Morgan recommends therefore that this formula be used for
the determination of molecular weights. After having calibrated

* Jour. Am. Chem. Soc., 30, 360 (1908); 30, 1055 (1908).

150 THEORETICAL CHEMISTRY

a particular tip with pure benzene (a liquid which is known to be
non-associated), and thus ascertaining the value of fc, the drop-
weights at several different temperatures are determined. If
we assume M to have the same value in the liquid and gaseous
states, the value of lc can be computed by substituting the experi-
mental data in the preceding equation. If at the different tem-
peratures at which drop-weights are determined, the same value
of tc is obtained, then we may infer that the liquid is non-asso-
ciated and, therefore, that the assumption made as to the value of
M is confirmed. It is a singular fact that the calculated value
of tc for some liquids does not agree with the experimental value,
although it remains constant throughout an extended range of
temperatures. Morgan considers a constant value of tc to be an
indication of non-association, even if the value is fictitious. In
this method the constancy of the calculated value of the critical
temperature becomes the criterion of molecular association, and
thus affords a means of determining whether the molecular weight
in the liquid state is identical with that in the gaseous state. The
values of tc calculated from the drop-weights of an associated
liquid become steadily smaller as the temperature increases. A
large number of liquids have been studied by this method, and
the results indicate that many of the substances which were con-
sidered to be associated by Ramsay and Shields are in reality
non-associated; in fact, it appears from the work of Morgan that
association is much less common among liquids than has hitherto
been supposed.

Dielectric Constants. In 1837, Faraday discovered that the
attraction or repulsion between two electric charges varies with
the nature of the intervening medium or dielectric. If gi and q%
represent two charges which are separated by a distance r, the
force of attraction or repulsion, /, is given by the equation

where D is a specific property of the medium known as the dielectric
constant. The dielectric constant of air is taken as unity. Vari-
ous methods have been devised for the experimental determination

LIQUIDS

151

of the dielectric constant, but the scope of this book forbids
even a brief description of the apparatus or an outline of the
processes of measurement. For a description of these methods
the student is referred to any one of the more complete physico-
chemical laboratory manuals, or to the original communications
of Nernst * and Drude.f

The values of the dielectric constants for some of the more
common solvents are given in the accompanying table.

DIELECTRIC CONSTANTS AT 18° C.

Substance.

Hydrogen dioxide

92 8

Water
Formic acid

77 0
63.0

Methyl alcohol

33 7

Ethyl alcohol

25 9

Ammonia, liquid

22 0

Chloroform

5 0

Ether . .

4 4

Carbon disulphide

2.6

Benzene

2 3

The importance of this property of liquids will become more
apparent in subsequent chapters, especially in those devoted to,
electrochemistry.

PROBLEMS.

1. It is desired to compare the molecular volumes of alcohol and ether.
If the molecular volume of ether is determined at 20° C., at what temper-
ature must the molecular volume of alcohol be determined? The boiling
points of alcohol and ether are 78° and 35° respectively. Ans. 61° C.

2. A volume of 50 liters of air in passing through a liquid at 22° C.
causes the evaporation of 5 grams of substance, the molecular weight of
which is 100. What is the vapor pressure of the liquid in grams per
square centimeter? . Ans. 25.

* Zeit. phys. Chem., 14, 622 (1894).
t Ibid., 23, 267 (1897).

152 THEORETICAL CHEMISTRY

3. The boiling-point of ethyl propionate is 98°.7 C. and its heat of
vaporization is 77.1 calories. Calculate its molecular weight.

4 The heat of vaporization of liquid ammonia at its boiling-point,
under atmospheric pressure (— 33°.5 C.) is 341 calories. Is liquid am-
monia associated?

5. Calculate the molecular volume of ethyl butyrate. The molecular
volume determined by experiment is 149.1.

6. For propionic acid, d = 1.0158 and nD = 1.3953. Calculate the
molecular refraction by the formula of Lorenz-Lorentz and compare the
value so obtained with that derived from the atomic refractions of the
constituent elements.

7. The density of ether is 0.7208, of ethyl alcohol, 0.7935 and of a
mixture of ether and alcohol containing p per cent of the latter, 0.7389.
At 20° C. the refractive indices for sodium light are, for ether, 1.3536, for
alcohol, 1.3619, and for the mixture, 1.3572. Calculate the value of p,
using the Gladstone and Dale formula. A us. 20.81.

8. At 20° C. the density of chloroform is 1.4823 and the refractive
index for the D-line is 1.4472. Given the atomic refractivities of carbon
and hydrogen, calculate that of chlorine, using the Lorenz-Lorentz
formula. Ans. 5.999.

9. Calculate the surface tension of benzene in dynes per centimeter,
the radius of the capillary tube being 0.01843 cm., the density of the
liquid, 0.85, and the height to which it rises in the capillary, 3.213 cm.

Ans. 24.71 dynes /cm.

10. Find the molecular weight of benzene, the surface tension at
46° C. being 24.71 dynes per centimeter, its critical temperature, 288°.5 C,,
its density, 0.85 and the value of k = 2.12. Ans. 77.7.

11. At 14°.8 C. acetyl chloride (density = 1.124) ascends to a height
of 3.28 cm. in a capillary tube the radius of which is 0.01425 cm. At
46°.2 C. in the same tube the elevation is 2.85 cm. and the density =
1.064. Calculate the critical temperature of acetyl chloride.

Ans. 234°.6C.

12. From a certain tip the weights of a falling drop of benzene are
35.329 milligrams (temp. = 11°.4, density = 0.888) and 26.530 milli-
grams (temp. = 68°.5, density = 0.827). The molecular weight is the
same in the liquid and gaseous states. Calculate the critical temper-
ature of benzene. Ans. 286M C.

CHAPTER VIII.
SOLIDS.

General Properties of Solids. Solids differ from gases and
liquids in possessing definite, individual forms. Matter in the
solid state is capable of resisting considerable shearing and tensile
stresses. In terms of the kinetic theory of matter, the mutual
attractive forces exerted by the molecules of solids must be re-
garded as superior to the attractive forces between the molecules
of gases and liquids. With one or two exceptions all solids ex-
pand when heated, but there is no simple law expressing the relation
between the increment of volume and the temperature. Rigidity is
another characteristic property of solids, it being much more ap-
parent in some than in others. Many solids are constantly under-
going a process of transformation into the gaseous state at their
free surfaces, such a change being known as sublimation. Just
as when a gas is sufficiently cooled it passes into the liquid state,
so on cooling a liquid below a certain temperature, it passes into
the solid state. The reverse transformations are also possible, a
solid being liquefied when sufficiently heated, and the resulting
liquid completely vaporized if the heating be continued. Heat
energy is required to effect transition from the solid to the liquid
state, just as heat energy is required to effect transition from the
liquid to the gaseous state.

Obviously a substance in the solid state contains less energy
than it does in the liquid state. The number of calories required
to melt 1 gram of a solid substance is called its heat of fusion.
It is often difficult to decide whether a substance should be classi-
fied as a solid or as a liquid. For example the behavior of certain
amorphous substances such as pitch, amber and glass, is similar
to that of a very viscous, inelastic liquid. Solids are generally
classified as crystalline or amorphous. In crystalline solids the
molecules are supposed to be arranged in some definite order, this

153

154 THEORETICAL CHEMISTRY

arrangement manifesting itself in the crystal form. An amor-
phous solid on the other hand may be considered as a liquid
possessing great viscosity and small elasticity. The • physical
properties of amorphous solids have the same values in all direc-
tions, whereas in crystalline solids the values of these properties
may be different in different directions. When an amorphous
solid is heated it gradually softens and eventually acquires the
properties characteristic of a liquid, but during the process of heat-
ing there is no definite point of transition from the solid to the
liquid state. On the other hand, when a crystalline solid is
heated there is a sharp change from one state to the other at a
definite temperature, this temperature being termed the melting-
point.

Crystallography. The study of the definite geometrical forms
assumed by crystalline solids is termed crystallography. The
number of crystalline forms known is exceedingly large, but it is
possible to reduce the many varieties to a few classes or systems
by referring their principal elements — the planes — to definite
lines called axes. These axes are so drawn within the crystal that
the crystal surfaces are symmetrically arranged about them.
This system of classification was proposed by Weiss in 1809.
He showed that notwithstanding the multiplicity of crystal forms
encountered in nature, it is possible to consider them as belonging
to one of six systems of crystallization.

The six systems of Weiss are as follows: —

1. The Regular System. Three axes of equal length, inter-
secting each other at right angles (Fig. 41).

2. The Tetragonal System. Two axes of equal length and the
third axis either longer or shorter, all three axes intersecting at
right angles (Fig. 42).

3. The Hexagonal System. Three axes of equal length, all in
the same plane and intersecting at angles of 60°, and a fourth axis,
either longer or shorter and perpendicular to the plane of the
other three (Fig. 43).

4. The Rhombic System. Three axes of unequal length, all
intersecting each other at right angles (Fig. 44).

SOLIDS

155

5. The Monodinic System. Three axes of unequal length, two
of which intersect at right angles, while the third axis is per-
pendicular to one and not to the other (Fig. 45).

6. The Tridinic System. Three axes of unequal length no two
of which intersect at right angles (Fig. 46).

The position of a plane in space is determined by three points
in a system of coordinates, and consequently the position of the

a
Fig. 41.

Fig. 42.

Fig. 44.

Fig. 45.

Fig. 46.

face of a crystal is likewise determined by its points of inter-
section with the three axes, or by the distances from the origin
of the system of coordinates at which the plane of the crystal
face intersects the three axes. These distances are called the
parameters of the plane.
The fundamental law of crystallography discovered by Steno

156 THEORETICAL CHEMISTRY

in 1669 may be stated thus: — The angle between two given crystal
faces is always the same for the same substance. The fact that
every crystalline substance is characterized by a constant inter-
facial angle, affords a valuable means of identification which is
used by both chemists and mineralogists. The instrument em-
ployed for the measurement of the interfacial angles of crystals is
called a goniometer. The crystal to be measured is mounted at
the center of the graduated circular table of the goniometer, and
the image of an illuminated slit, reflected from one surface of the
crystal, is brought into coincidence with the cross-wires in the eye-
'piece of the telescope. The table is then turned until the image
of the slit, reflected from the adjacent face of the crystal, coincides
with the cross-wires. The interfacial angle of the crystal is de-
termined by the number of degrees through which the table has
been turned.

Properties of Crystals. The properties of all crystals, except
those belonging to the regular system, exhibit differences, depend-
ent upon the direction in which the particular moasurernents
are made. Thus the elasticity, the thermal and electrical con-
ductivities, and in fact all of the physical properties of crystals
which do not belong to the regular system, have different values
in different directions. Crystals whose physical properties have
the same values in all directions are termed isotropic, while those
in which the values are dependent upon the direction in which
the measurements are made, are called anisotropic. Crystals
belonging to the regular system, and amorphous substances are
isotropic. Certain amorphous substances, such as glass, which
are normally isotropic, may become anisotropic when subjected
to tension or compression. The phenomenon of double refrac-
tion observed in all crystals, except those belonging to the regular
system, is due to their anisotropic character. Crystals belonging
to the tetragonal and hexagonal systems resemble each other in
one respect, viz. : that in all of them there is one direction, called
the optic axis, or axis of double refraction (coincident with the prin-
cipal crystallographic axis), along which a ray of light is singly
refracted, while in all other directions it is doubly refracted. In
crystals belonging to the rhombic, monoclinic, and triclinic systems,

SOLIDS 157

there are always two directions along which a ray of light is singly
refracted. A crystal of Iceland spar (CaC03) affords a beauti-
ful illustration of double refraction. On placing a rhomb of this
substance over a piece of white paper on which there is an ink
spot, two spots will be seen. On turning the crystal, one spot will
remain stationary while the other spot will revolve about it. This
property of Iceland spar is utilized in the construction of Nicol
prisms for polariscopes.

The examination of sections of anisotropic crystals in a polari-
scope between crossed Nicol prisms, reveals something as to their
crystal form. As has been stated, crystals of the tetragonal and
hexagonal systems are uniaxiaL If a section is cut from such a
crystal perpendicular to the optic axis, and this is placed between
the crossed Nicol prisms of a polariscope, in a convergent beam
of white light, a dark cross and concentric, spectral-colored circles
will be observed, Fig. 47. Upon turning the analyzer through
90° the colors of the circles will change to the respective comple-
mentary colors and the dark cross will become light. Crystals
of the rhombic, monoclinic, and triclinic systems are biaxial. If
a section of a biaxial crystal, cut perpendicular to the line bisect-
ing the angle between the two axes, be placed in. the polatiscope

Fig. 46. Fig. 47.

and examined as in the preceding case, a series of concentric
spectral-colored lemniscates surrounding two dark centers and
pierced by dark, hyperbolic brushes, will be observed, as shown in
Fig 48. On rotating the analyzer, the colors will change to the
corresponding complementary colors, as in the case of uniaxial
crystals. The appearance of these figures is so varied and char-
acteristic as to furnish, in many cases, a very satisfactory means
of identifying anisotropic crystals.

158 THEORETICAL CHEMISTRY

Etch Figures. The solubility of crystals has been shown to
be different in different directions. Thus, if the surface of a crys-
talline substance be highly polished and then treated for a short
time with a suitable solvent, faint patterns, known as etch figures,
will appear as a result of the inequality of the rate of solution, in
different directions. When these figures are examined under the
microscope the crystal-form can generally be determined. The
examination of etch figures has come to be of prime importance
to the metallographer. Thus, when an appropriate solvent is
applied to the polished surface of an alloy, not only is the crystal
form revealed by the etch figures, but also the presence of various
chemical compounds may be recognized. By a careful study
of the etch figures developed on the surface of highly polished
steel, the metallographer may gather important information as to
its previous history, especially its heat treatment.

Crystal Form and Chemical Composition. From the pre-
ceding paragraphs it might be inferred that the same substance
always assumes the same crystal form. While this is true
in general, there are some substances which appear in several
different crystal forms. This phenomenon is termed polymor-
phism.

Calcium carbonate is an example of a substance crystallizing in
more than one form. As calcite, it crystallizes in the hexagonal
system, while as aragonite, it crystallizes in the rhombic system.
Such a substance is said to be dimorphous. Of the several factors
controlling polymorphism, temperature is the most important.
Thus sulphur crystallizes at temperatures above 95°.6 in the mon-
oclinic system, while at lower temperatures it assumes the
rhombic form. The temperature at which it changes from one
form into the other is termed its transition temperature. As has
been mentioned in an earlier chapter (p. 14), some substances
may crystallize in the same form, the characteristic interfacial
angles being nearly identical. Such substances are said to be iso-
morphous. This phenomenon, discovered by Mitscherlich, has
been of great use in connection with the earlier investigations on
atomic weights, as has already been pointed out.

There can be little doubt as to the existence of an intimate

SOLIDS 139

connection between crystalline form and chemical composition.
Ever since the early part of the nineteenth century, when Hauy
established the science of crystallography, various attempts have
been made by chemists and crystallographers to connect crystal-
line form with chemical constitution. In 1906, Barlow and Pope *
made a most notable contribution to the theories concerning the
relation between crystalline form and chemical constitution.
Their ideas may be summarized as follows: — If each atom be
considered as appropriating a certain space, called its sphere of
atomic influence, then (1) the spheres of atomic influence are so
arranged as to occupy the smallest possible volume in every crystal;
(2) the volumes of the spheres of atomic influences in any substance
are proportional to the valences of the constituent atoms; (3) the
volumes of the spheres of influence of the atoms of different elements
of the same valence are nearly equal, any variation being in har-
mony with their relations in the periodic system. Barlow and Pope
have shown that the general agreement between theory and
observation is most satisfactory, a particularly strong argument
in favor of this theory being the very plausible explanation which
it furnishes for a large number of crystallographic facts. It is
without doubt the best working hypothesis which has yet been
offered for the investigation of the dependence of crystalline form
upon a definite chemical constitution.

Compressibilities of the Solid Elements. A series of careful
measurements of the compressibilities of the elements by T. W.
Richards and his collaborators^ has revealed the fact that com-
prefosibility is a periodic function of atomic weight. Richards
has advanced some interesting suggestions as to the importance
of compressibility in connection with intermodular cohesion
and atomjjp volume. In fact, Richards' theory of compressible
atoms may be regarded as a valuable supplement to the the-
ory of Barlow and Pope and, taken together, these two theories
constitute a rational basis for the science of chemical crystallog-
raphy.

X-Rays and Crystal Structure. In 191$fllaue pointed out

* Jour. Chem. Soc., pi, 1150 (1907).

t Zeit. phys. Chein'., 6x, 77, 100, 171, 183 (1008),

160 THEORETICAL CHEMISTRY

that the regularly arranged atoms or molecules of a crystal should
act as a three-dimensional diffraction grating toward the X-rays.
He showed mathematically that on traversing a thin section of a
crystal, a pencil of X-rays should give rise to a diffraction pattern
arranged symmetrically round the primary beam as a center. A
photographic plate placed perpendicular to the path of the rays
and behind the crystal should reveal, on development, a central
spot due to the action of the primary rays, and a series of symmetri-
cally grouped spots due to the diffracted rays.

Laue's predictions were verified experimentally by Friedrich
and Knipping, who obtained numerous plates showing a va-
riety of geometrical patterns corresponding to the structural dif-
ferences of the crystals examined. The analysis of the Laue
diffraction patterns, while furnishing valuable information as to
the internal structure of crystals, is nevertheless extremely
complex.

W. H. Bragg and his son W. L. Bragg * have devised an X-ray
spectrometer in which use is made of the fact that the regularly
spaced atoms of a crystal reflect the X-rays in much the same way
that light is reflected (diffracted) by a plane grating. By observing
the angles of reflection from the different faces of a crystal for an
incident radiation of known wave-length, it is an easy matter to
calculate the distances between the atoms of the crystal which
function as diffraction centers.

By means of the X-ray spectrometer the internal structure of a
number of crystals has been determined. One of the most in-
teresting results to the chemist is that obtained with the diamond.
The X-ray spectra of the diamond reveal the fact that each carbon
atom is situated at the center of a regular tetrahedron formed by
four other carbon atoms.

In commenting on this method of studying crystal structure,
W, H. Bragg says: — " Instead of guessing the internal arrange-
ment of the atomtJrom the outward form assumed by the crystal,
we find ourselvift^rae to measure the actual distances from atom
to atom and to oMw a diagram as if we were making a plan of a
building."

* See "X-Rays and Crystal Structure," by W. H. Bragg and W. L. Bragg.

SOLIDS 161

Heat Capacity of Solids. Recent investigations of s
heats of solids at extremely low temperatures have resulted in
the formulation of several interesting relationships between heat
capacity and temperature.

At ordinary temperatures the molecules of a crystalline solid
may be assumed to be in a state of violent, unordered motion. As
the temperature is lowered, the amplitude of the molecular oscil-
lations steadily diminishes until finally, at the absolute zero, there
is in all probability a complete cessation of motion. In the neigh-
borhood of the absolute zero, where the amplitude of the molecular
oscillations is negligible, a crystalline solid may be assumed to
possess the properties characteristic of a perfectly elastic body.
In other wordtf, the crystalline forces holding the molecules to-
gether would preponderate over the feeble thermal forces tending
to initiate molecular oscillations within the solid. Under these
conditions the solid as a whole would exhibit the same behavior as
a single molecule, that is to say, the solid would function as a
perfectly elastic body.

pn this assumption Debye * has derived the following equation
expressing the heat capacity of a solid, Cv, in terms of its absolute
temperature T,

In this equation 0 is a constant characteristic of each solid and has
the same dimensions as T. The value of 6 varies between the
lllits 6 = 50 for calcium and 6 = 1840 for carbon. The agree-
ment between the observed and calculated values of Cv has been
found to be excellent up to T = 6/12.

Whea^us latter temperature is exceeded, the molecules of the
solid begm to absorb more and more heat energy and to vibrate
independently about their centers of oscillation. The failure of
Debye's equation is to be expected under these conditions since
the solid is no longer behaving as one large||ji&eule. Obviously
the lighter the molecules and the greater "^P'crystalline forces
. within the solid, the higher must the temperature become before

* Ami. Physik., 39* 789 (1912).

162 THEORETICAL CHEMISTRY

the individual molecules can acquire appreciable kinetic energy.
This is apparent from the familiar dynamical principle, that the
kinetic energy of a vibrating particle is proportional to its mass
and to the square of its vibration frequency. In the case of lead,
which is a soft, malleable solid with a relatively low melting-point,
it is reasonable to infer that the crystalline forces are feeble, and
consequently we should expect that molecular and atomic vibra-
tions would be set up at quite low temperatures. Furthermore,
since the atoms of lead are extremely heavy, their kinetic energy
must be great. The correctness of these conclusions is confirmed
by the fact that the Debye equation when applied to lead has been
found to hold only over a very short range _of temperature. On
the other hand, the equation has. beeif found to hold for the
diamond up to a temperature of about 'SOO9 absolute. In this
case we have a solid in which the crystalline forces are extremely
powerful and in which the atoms are relatively light. A fairly
high temperature must be attained before the energy absorbed
by the individual atoms of the diamond acquires appreciable
magnitude.

The absorption of energy by the vibrating molecules continues
to increase as the temperature is raised until ultimately, at the
melting point of the solid, the crystalline forces become negligible.
As this temperature is approached therefore, the intermolecu-
lar restraint becomes less and less and the mean kinetic energy
of the molecules approaches that of the molecules of the molten
solid.

As has already been stated in Chapter I (p. 11), Dulong and
Petit, in 1819, discovered the interesting fact that the atomic heats
of the solid elements have a constant value of 6.5. The importance*
of this generalization in connection with the verification of atomic
weights has already been pointed out. Quite recently, Lewis *
has directed attention to the fact that it is much more rational to
calculate the atomic heat of an element from the specific heat at
constant volume rather than from the specific heat at constant
pressure. While it is impossible to measure the specific heat at
constant volume, its value may be derived from the specific heat
* Jour. Am. Chem. Soc,, 29, 1165 (1907).

SOLIDS 163

at constant pressure by an application of the laws of thermo-
dynamics. Thus, Lewis has obtained the formula

c - c - ^!H

°" Uv~ 41.78/3'

where T denotes the absolute temperature, a the coefficient of
expansion, 0 the coefficient of compressibility, Cp and Cv the
atomic specific heats at constant pressure and constant volume
respectively and V the atomic volume. By means of this equa-
tion, Lewis has established the following generalization: Within
the limits of experimental error, the atomic heat at constant volume,
at 20° C., is the same for all solid elements whose atomic weights are
greater than that of potassium, and is equal to 5.9. In the case of a
solid having a high melting-point, the violent agitation of its con-
stituent molecules and atoms as the temperature is raised, will
undoubtedly produce a corresponding increase in the amplitude of
vibration of its electrons together with an increase in their trans-
lational velocity among the molecules. Under these conditions,
the specific heat at constant volume should be greater than 5.9
calories. This conclusion cannot be verified experimentally until
the values of a and /3 in the foregoing equation have been deter-
mined at high temperatures.

The complete heat capacity curves for three typical solid ele-
ments, lead, aluminium, and carbon, are given in Fig. 49. It is
apparent from these curves that the absorption of heat energy by
a crystalline solid may be considered as taking place in three dis-
tinct stages, as follows: — (1) In the neighborhood of the abso-
lute zero, the heat capacity remains practically zero; (2) the heat
capacity increases rapidly with the temperature; and (3) the heat
capacity increases slowly, approaching asymptotically the limiting
value 5.9 for T = ». For a malleable, low melting element of
high atomic weight, such as lead, the first two stages are very
short and the final stage commences at a low temperature. On
the contrary, with a hard, high melting element of low atomic
weight, such as carbon in the form of diamond, the final stage is
not reached at any temperature within the range covered by the
experiments.

164

THEORETICAL CHEMISTRY

The Nernst-Lindemann Equation. Recently, several equa-
tions have been derived expressing the heat capacity of a solid in
terms of temperature and arbitrary constants. All of these
equations are based upon the so-called "quantum theory" accord-
ing to which the absorption of heat energy by matter is supposed
to take place in a discontinuous manner, the discrete units of
energy being termed quanta. While the discussion of the quantum
theory and the equations connecting heat capacity and tempera-
ture lies outside of the scope of this book, mention should never-
theless be made of the empirical equation derived by Nernst and
Lindemann.* This equation gives values of Cv which are in re-

500

markably close agreement with the values determined by direct
experiment. The equation may be written in the following form : —

c,=

'-I-'

(eT — I)2 (e2 T — 1)

In this expression, R is the molecular gas constant R = 2 calories,
6 is the base of the natural system of logarithms and 0 is a constant
depending upon the nature of the solid.

* Zeit. Elektrochem., 17, 817 (1911).

SOLIDS 165

The value of 0 may be calculated with a fair degree of accuracy
by means of the equation

in which T/ denotes the absolute melting-point of the substance,
A its atomic weight and d its density.

Liquid Crystals. In addition to possessing well-defined geo-
metrical forms, crystalline substances are characterized by their
resistance to deformation when subjected to mechanical stress,
and by the property of melting sharply at definite temperatures
with the production of transparent liquids.

In 1888, two substances, cholesteryl acetate and cholesteryl
benzoate, were found by Reinitzer * to behave in an anomalous
manner when heated. At definite temperatures these substances
melted to turbid liquids which, in turn, became clear on further
heating, the latter change also taking place at definite tempera-
tures. On cooling the clear liquids, the reverse series of changes
was found to occur.

Examination of the turbid liquids revealed the fact that they
resembled ordinary liquids in their general behavior, such as assum-
ing the spherical shape when suspended in a medium of the same
density, or of rising in a capillary tube under the influence of sur-
face tension. But in addition to possessing the properties char-
acteristic of the liquid state, Lehmann discovered that they
possessed optical properties which had hitherto been observed
only with solid, crystalline substances. Their behavior towards
polarized light was such as to warrant the conclusion that these
turbid liquids are anisotropic. In view of these facts, Lehmann
proposed that liquids possessing these properties should be called
liquid crystals, the term implying that under ordinary condi-
tions, the crystalline forces in these substances are so feeble that
the crystals readily undergo deformation and actually flow like
liquids. That these turbid liquids are not emulsions, is proven
by the fact that when they are examined under the micro-
scope, the turbidity is found to be due to the aggregation of a

* Monatshefte, 9, 435 (1888).

166 THEORETICAL CHEMISTRY

myriad of differently oriented transparent crystals. All subse-
quent investigation of liquid crystals has failed to show any lack
of homogeneity.

The number of such substances known at the present time is
fairly large.

CHAPTER IX.
SOLUTIONS.

Classification of Solutions. Having dealt with the properties
of pure substances in the gaseous, liquid and solid states we now
proceed to the consideration of the properties of mixtures of two
or more pure substances. When such a mixture is chemically
and physically homogeneous, and no abrupt change in its prop-
erties results from an alteration of the proportions of the com-
ponents of the mixture, it is termed a solution. When one
substance is dissolved in another, it is customary to designate as
the solvent that component which is present in the larger proportion,
the other component being termed the solute. When not more
than one-tenth mol of solute is present in one liter of solution, the
solution is said to be dilute. The detailed study of dilute solu-
tions will be deferred until the next chapter.

There are nine possible classes of solutions, as follows: —

(1) Solution of gas in gas; v^

(2) Solution of liquid in gas; v

(3) Solution of solid in gas; >

(4) Solution of gas in liquid; w^

(5) Solution of liquid in liquid;

(6) Solution of solid in liquid; ^

(7) Solution of gas in solid;

(8) Solution of liquid in solid;

(9) Solution of solid in solid.

While examples of all of these different types of solutions are
known, only the more important classes will be considered here.

Solutions of Gases in Gases. In solutions of this class the
components may be present in any proportions, since gases are
completely miscible. In a mixture of gases where no chemical
action occurs, each gas behaves independently, the properties of

167

168

THEORETICAL CHEMISTRY

the gaseous mixture being the sum of the properties of the con-
stituents. Thus, the total pressure of a mixture of several gases is
equal to the sum of the pressures which each gas would exert were it
alone present- in the volume occupied by the mixture. This law was
discovered by Dalton * and is known as Dalton's law of partial
pressures. If the partial pressures of the constituent gases be
denoted by pi, p2, PS, etc., and P and V represent the total pres-
sure and the total volume of the gaseous mixture, then

PV = V (pi + p* + p3 +•••)•

Dalton's law holds when the partial pressures are not too great,

its order of validity being the same
as that of the other gas laws. Dai-
ton's law can be tested experimen-
tally by comparing the total pressure
of the gases with the sum of the pres-
sures exerted by each gas before
mixture. van't Hoff pointed out
the possibility of measuring the par-
tial pressure of one of the two com-
ponents of a gas mixture, provided
a diaphragm could be found which
would be pervious to one of the
gases but not to the other. It was
shown shortly afterward by Ram-
say,! that the walls of a vessel of
palladium, when sufficiently heated,
permit the free passage of hydrogen
but not of nitrogen. The walls are

Hydrogen

Fig. 50.

said to be semi-permeable. A sketch of the apparatus used by
Ramsay in the verification of Dalton's law is shown in Fig. 50.
A small vessel of palladium, P, containing nitrogen, is connected
with a manometer AB, which serves to measure the pressure of the
gas in P. The vessel P is enclosed within a larger vessel C, which
can be filled with hydrogen at known pressure. On heating P and

* Gilb. Ann., 12, 385 (1802).
t Phil. Mag. (5), 38, 206 (1894).

SOLUTIONS 169

passing a current of hydrogen at a definite pressure through C, the
hydrogen enters P until the pressures due to hydrogen inside and
outside are equal. The total pressure in P, measured on the
manometer, is greater than the pressure in C. The difference
between the two pressures is very nearly equal to the partial pres-
sure of the nitrogen. Conversely, if a mixture of the two gases
be introduced into P, which is then heated and maintained at suf-
ficiently high temperature to insure its permeability to hydrogen,
the partial pressure of the nitrogen can be determined by passing
a current of hydrogen at known pressure through C until equilib-
rium is attained, as shown by the manometer. The difference
between the external and internal pressures is the partial pressure
of the nitrogen. This experiment has a very important bearing
upon the modern theory of solution.

Solutions of Gases in Liquids. The solubility of gases in
liquids is limited, the extent to which they dissolve depending
upon the pressure, the temperature, the nature of the gas, and
the nature of the solvent. When a liquid cannot absorb any
more of a gas at a definite temperature, it is said to be saturated,
and the solution is called a saturated solution. The solubility of
a gas in a liquid is defined by Ostwald as the ratio of the volume
of the gas absorbed to the volume of the absorbing liquid at a
specified temperature and pressure, or if the solubility of the gas
be represented by S9 we have

S = vfV,

where v is the volume of gas absorbed and V is the volume of the
absorbing liquid. The " absorption coefficient" of Bunsen in
terms of which he expressed the results of his measurements of
the solubility of gases, may be defined as the volume of a gas,
reduced to 0° C. and 76 cm. pressure which is absorbed by unit
volume of a liquid at a certain temperature and under a pressure
of 76 cm. of mercury. In certain cases the volume of the gas
absorbed is found to be independent of the pressure, so that if a
is the coefficient of gaseous expansion, and ft Bunsen's coefficient
of absorption, then

S- 0(1 + «0.

170

THEORETICAL CHEMISTRY

The solubilities of a few gases in water and alcohol as determined
by Bunsen are given in the following table: —

Watei.

Alcohol.

Gas

0°

15°

0°

15°

Hydrogen

0 0215

0 0190

0 0693

0 0673

Oxygen . .

0 0489

0 0342

0 2337

0 2232

Carbon dioxide. . . .

1 797

0 1002

4 330

3 199

The solubility of gases in water is appreciably diminished by
the presence of dissolved solids or liquids, especially electrolytes.
Various theories have been proposed to account for the diminished
solubility of gases in salt solutions but the most satisfactory is
that due to Philip,* who suggests that the phenomenon is caused
by the hydration of the dissolved salt. A portion of the water
in the salt solution is supposed to be in combination with the salt,
the water which is thus removed from the role of solvent, being
no longer free to absorb gas. The solubility of a gas increases
with increase in pressure. For gases which do not react chem-
ically with the solvent, there exists a simple relation between
pressure and solubility, discovered by Henry.f This relation,
known as Henrys law may be stated as follows: — When a gas is
absorbed in a liquid, the weight dissolved is proportional to the pressure
of the gas. Since pressure and volume, at constant temperature,
are inversely proportional (Boyle's law), the law of Henry may be
stated thus: — The volume of a gas absorbed by a given volume of
liquid is independent of the pressure. There is yet another form
in which the law may be stated which is instructive in connection
with the modern theory of solution. When a definite volume of
liquid is saturated with a gas at constant temperature and pres-
sure, a condition of equilibrium is established between the gas in
solution and that in the free space over the solution, therefore,
Henry's law may be stated as follows: — The concentration of the
dissolved gas is directly proportional to that in the free space above

* Trans. Faraday Soc., 3, 140 (1907).
t Gilb. Ann., 20, 147 (1805).

SOLUTIONS 171

the liquid. If Ci represents the concentration of the gas in the
liquid and c% the concentration in the free space above the liquid,
Henry 's law may be expressed thus: Ci/c% = k,
where k is known as the solubility coefficient.

Dalton showed that the solubility of the individual gases in a
mixture of gases is directly proportional to their partial pressures,
the solubility of each gas being nearly independent of the presence
of the others.

As will be seen from the foregoing table, the solubility of a gas
in a liquid diminishes with increase in temperature.* Concerning
the influence of the nature of the gas on its solubility, it may be
said that those gases which exhibit acid or basic reactions are
the most soluble, the solubilities of neutral gases being small.
In the case of many of the very soluble ffasesJHenry's law does
riot hold. For example, ammonia, a gas having marked basic
properties and a large coefficient of solubility, does not obey
Henry's law at ordinary temperatures, the mass of ammonia
absorbed not being proportional to the pressure. The curve
showing the variation in solubility with pressure at 0° C. has
two marked discontinuities. At temperatures above 100° C.
the gas obeys Henry's law. Sulphur dioxide behaves similarly,
the law holding only for temperatures exceeding 40° C.

With regard to the connection between the solvent power of a
liquid and its nature but little is known. About all that can be
said is, that the order of aolubilitff_of gasejyn different J^uidsis^
the sam£._ Thus in the preceding table the solubilities of hydro*
gen, oxygen and carbon dioxide in water and in alcohol will be
seen to be approximately proportional. A slight ' change in
volume always results when a gas is dissolved in a liquid. In
general it may be said that the less compressible a gas is, the
greater is the increase in volume produced when it is absorbed by
a liquid. It is of interest to note that the increase in volume
caused by the solution of a gas is nearly equal to the value of b
for the gas in the equation of van der Waals. This is shown m
the following table: —

* Helium is an exception to the rule that the solubility of a gas in a liquid
diminishes with increase in temperature. The absorption coefficient of He
diminishes from 0° to 25° and then increases again as the temperature is raised.

172 THEORETICAL CHEMISTRY

Gas.

Increase in Vol.

Oxygen ...

0 00115

0 000890

Nitrogen

0 00145

0 001359

Hydrogen ...

0 00106

0 000887

Carbon dioxide

0 00125

0 OQ0866

solutions of Liquids in Liquids. Solutions of liquids in liquids
can be divided into three classes as follows: — (1) Liquids which
are miscible in all proportions; (2) Liquids which are partially
miscible; and (3) Liquids which are immiscible. Examples of
these three classes in the order mentioned are, alcohol and water,
ether and water, and benzene and water. As to the cause of
miscibility and non-miscibility of liquids very little is known.

Partial Miscibility. If a small amount of ether is added to
a large volume of water in a separatory funnel and the mixture
vigorously shaken, a perfectly homogeneous solution will be
obtained. On gradually increasing the amount of ether, shaking
after each addition, a concentration will eventually be reached
at which a separation into two layers will take place. The upper
layer is a saturated solution of water in ether and the lower layer
is a saturated solution of ether in water. So long as the relative
amounts of the two liquids is such that the mixture does not
become homogeneous on standing, the composition of the two
layers will be independent of the relative amounts of the two
components. Measurements of the mutual solubility of liquids
have been made by Alexieeff * by placing weighed amounts in
sealed tubes and observing the temperature at which the mixture
became homogeneous. In general the solubility of a pair of
partially miscible liquids increases with the temperature, and
therefore it may be inferred that at a sufficiently high temperature
the mixture will become perfectly homogeneous. An example of
this type of binary mixture is furnished by phenol and water, the
solubility curve of which is shown in Fig. 51. In this diagram
temperature is plotted on the axis of ordinates and percentage
composition of the solution on the axis of abscissae. Starting

* Jour, prakt. Chem., 133, 518 (1882); Bull. Soc. Chem., 38, 145 (1882).

SOLUTIONS

173

with a small amount of phenol and adding it in increasing quan-
tities to a large volume of water, a concentration will eventually
be reached at which the solution will separate into two layers.
This concentration is represented by the point A. On raising the

100*

Percentage Water in Phenol

0 i Percentage Phenolln Water

Fig. 51.

100*

temperature, the solubility of phenol in water increases, as shown
by the curve AB. In like manner, starting with pure phenol
and adding increasing amounts of water, separation into two layers
will occur at a concentration represented by the point C. As the
temperature is raised the solubility of water in phenol increases,
as shown by the curve CB. When the temperature is raised
above 68°.4 C., corresponding to the point B, phenol and water
become miscible in all proportions.

If we start with a solution whose temperature and composition
is represented by the point a, the addition of increasing amounts
of phenol, at constant temperature will be represented by the
dotted line afed. When the point / is reached, the solution will
separate into two layers the composition of which will be inde-
pendent of the relative amounts of phenol and water. At e the
solution will again become homogeneous. If the solution repre-

174 THEORETICAL CHEMISTRY

sented by the point a be again chosen as the starting point, and its
composition be kept unaltered while the temperature is raised to a
value above 68°. 4 C., the change will be represented by the dotted
line ab. If now the temperature be maintained constant and the
percentage of phenol increased, the alteration in composition will
be effected without discontinuity, as represented by the dotted
line be. On cooling the solution represented by the point c to the
initial temperature of a, the point d will be reached. Thus it is
possible to pass from a to d by the path abed without causing a
separation of the components into two layers. There is an
analogy between the solubility curve of a pair of partially mis-
cible liquids and the dotted, parabolic curve in the diagram of
the isothermals of carbon dioxide, shown in Fig. 23. In both
cases there is but one phase outside of the curves, while two
phases are coexistent within the area enclosed by the curves. The
analogy may be traced further, since in each case only one phase
can exist above a certain temperature. The temperature corre-
sponding to the apex of the parabolic curve in Fig. 23, is termed
the critical temperature of carbon dioxide and by analogy the
temperature corresponding to the point, /?, in Fig. 51 is called the
critical solution temperature. The mutual solubilities of some
pairs of partially miscible liquids were found by Alexieeff to di-
minish with increasing temperature. Thus a mixture of ether and
water, which is perfectly homogeneous at ordinary temperatures,
becomes turbid on warming. A specially interesting pair of
liquids is nicotine and water. At ordinary temperatures these
liquids are miscible in all proportions. If the temperature is
raised above 60° C., the solution becomes turbid owing to incom-
plete miscibility. On continuing to heat the mixture the
mutual solubility of the liquids begins to increase, until at
210° C. they become completely soluble again. The solubility
relations of this binary mixture are shown in Fig. 52. The closed
solubility curve defines the limits of the coexistence of two layers,
all points outside of the curve representing homogeneous solu-
tions.

Complete Miscibility: The study of the vapor pressures of
binary mixtures of completely miscible liquids is of great im-

SOLUTIONS

175

portance in connection with the possibility of separating them
by the process of distillation. The experimental investigations of
Konowalow * on homogeneous binary mixtures of liquids have
shown that such pairs of liquids may be divided into three classes

100*

Percentage Water in Nicotine

810°

60'

0 £ Percentage Nicotine in Water

Fig, 52.

100*

as follows: — (1) Mixtures having a maximum vapor pressure
corresponding to a certain composition, e.g., propyl alcohol and
water; (2) Mixtures having a minimum vapor pressure corre-
sponding to a certain composition, e.g., formic acid and water,
and (3) Mixtures having vapor pressures intermediate between
the vapor pressures of the pure components, e.g., methyl alcohol
and water. In considering the possibility of separating binary
mixtures of liquids belonging to these three classes, it is essential
to determine the composition of both solution and escaping vapor.
When a pure liquid is boiled the composition of the escaping
vapor is the same as that of the liquid itself, but this is, in general,

* Wied. Ann., 14, 34 (1881).

176

THEORETICAL CHEMISTRY

not the case when a binary mixture is distilled. The composition
of the liquid mixture in the distilling flask generally alters contin-
uously when such a mixture is distilled.

(1) The relation between the vapor pressure and composition
of all possible mixtures of propyl alcohol and water is represented
graphically in Fig. 53. In this diagram the compositions of the
mixtures are plotted as abscissae and vapor pressures as ordinates.
The vapor pressures of the pure components, water and propyl
alcohol, at a definite temperature are represented by A and C.
The maximum in the yapor-pressurecurve corresponds to a mix-

100*

Water

Propjl Alcohol
Fig. 53.

80*

ture containing 80 per cent of propyl alcohol. The dotted curve
represents the boiling-points of the various mixtures under normal
atmospheric pressure. Konowalow has shown^jthat the vapor of
abinary mixture)with JL minimum or maximum bqUing-pointfhas
t^^m^jeomposition as_that of the liquid.) The vagor^of all
mixtures containing less than 80 per cent of propyl alcohol will
be relatively richer in alcohol than the liquid mixture, since the
vapor of propyl alcohol is quite insoluble in water. If the amount
of alcohol in the mixture exceeds 80 per cent, then the vapor will
be relatively richer in water. Thus, whatever may be the com-
position of the mixture in the distilling flask, the distillate will
approximate to the composition of the mixture having the mini-
mum boiling-point. The residue in the flask will gradually

SOLUTIONS

177

change to pure water if the original concentration were below
80 per cent, or to pure alcohol if the original concentration were
above 80 per cent.

(2) The second type of binary mixture of liquids is illustrated
by formic acid and water, the vapor pressure and boiling-point
curves for which are shown in Fig. 54. A mixture containing
73 per cent of formic acid has a minimum vapor pressure and a
maximum boiling-point. At this concentration the vapor and the
liquid have the same composition. The vapor of mixtures con-

100*

Water

formic Acid*
Fig. 54.

taining less than 73 per cent of acid is relatively richer in water
than the liquid, while the vapor of mixtures containing more than
73 per cent of acid contains relatively less water than the liquid.
Any mixture of formic acid and water when distilled will thus
leave a residue in the distilling flask containing 73 per cent of
acid; this residue will distil at constant temperature like a homo-
geneous liquid. It was thought for a long time that such constant
boiling mixtures were definite chemical compounds of the two
liquids. Thus a mixture of hydrochloric acid and water contain
ing 20.2 per cent of acid boils at 110° C. under atmospheric pres-
sure. The composition of such a mixture corresponds very nearly
to the formula, HC1.8 H2O. Roscoe * showed that these mix-

* Lieb. Ann., 116, 203 (1860).

178

THEORETICAL CHEMISTRY

tures are not definite chemical compounds since the composition
of the distillate changes when the distillation is carried out under
different pressures.

(3) The vapor-pressure and boiling-point curves for methyl
alcohol and water, a mixture typical of the third class of com-
pletely miscible liquids, are shown in Fig. 55, the heavy line
representing vapor pressures at 65°.2 C. and the dotted line the
boiling points under normal, atmospheric pressure. In this case

oo*

Water

OH Methyl Alcohol 100*

Fig. 55.

the composition of both vapor and liquid alter continuously on
distillation. The distillate will contain a relatively larger amount
of alcohol and the residue in the distilling flask, an excess of water.
If this distillate be redistilled from a clean flask, a second dis-
tillate still richer in alcohol will be obtained. By repeating this
process a sufficient number of times, a more or less complete
separation of the two components of the mixture can be effected.
This process is termed fractional distillation.

Immistibility. When two immiscible liquids are brought
together, the total vapor pressure is equal to the sum of the vapor
pressures of the components; hence when such a mixture is dis-
tilled, the two liquids will pass over in the ratio of their respective

SOLUTIONS 179

vapor pressures, the boiling-point of the mixture being the temper-
ature at which the sum of the vapor pressures of the two liquids
is equal to the pressure of the atmosphere. The relation between
vapor pressure and composition in this case will be represented
by a horizontal line drawn at a distance above the axis of abscissae
equal to the sum of the vapor pressures of the components.

Nitrobenzene and water may be chosen as an example of a pair
of liquids which are practically immiscible. Under a pressure
of 760 mm. the mixture boils at 99° C. The vapor pressure of
water at this temperature is 733 mm.; the vapor pressure of nitro-
benzene muSt be 760 — 733 = 27 mm. Notwithstanding the
relatively small vapor pressure of nitrobenzene in the mixture,
considerable quantities of it distil over with the water. It is this
fact that makes possible separations of liquids by the process of
steam distillation so frequently employed by the organic chem-
ist. The relative weights of water and nitrobenzene passing
over in a steam distillation may be calculated as follows: — The
relative volumes of steam and vapor of nitrobenzene which distil
over will be in the ratio of their respective vapor pressures at the
temperature of the experiment, and consequently the relative
weights of the two liquids which pass over will be in the ratio,
p\di : ptfk, where pi and p2 denote the respective vapor pressures
of water and nitrobenzene, and d\ and d2 the corresponding vapor
densities. If w\ and w2 denote the weights of the two liquids in
the state of vapor, then

or, since vapor density is proportional to molecular weight, we
may write

Substituting in this proportion the values given above for the vapor
pressures of steam and nitrobenzene, we have

MI : w2 :: 733 X 18 : 27 X 123
or,

Wi : m :: 13,194 : 3321.

Thus the weights of water and nitrobenzene in the distillate are
approximately in the ratio of 4 to 1 notwithstanding the fact that

180

THEORETICAL CHEMISTRY

the ratio of their vapor pressures at the boiling-point of the mix-
ture is 27 to 1. If an organic substance is not decomposed by
steam, it is possible to effect an appreciable purificationj>y steam
distillation, even though its vapor pressure be relatively small.
As will be seen from the above example, it is the

weight^ of the nitrobenzene which compensates for its low vapor
pressure. It is the small molecular weight ,of_ water which renders
it so suitable for steam distillation.

Finally, the vapor-pressure and boiling-point relations of binary
mixtures of partially miscible liquids must be considered. In

100*

Water

700 mm.-

Isobutyl Alcohol
Fig. 56.

100*

general when two liquids are mixed, each lowers the vapor pressure
of the other, so that the vapor pressure of the mixture is less than
the sum of the vapor pressures of the components. As has al-
ready been pointed out, the composition of the two layers in a
binary mixture of partially miscible liquids is independent of
the relative amounts of the components present; hence the vapor
pressure remains constant so long as the solution remains hetero-
geneous. The vapor-pressure and boiling-point curves for a
binary mixture of partially miscible liquids (isobutyl alcohol and
water) are shown in Fig. 56. The horizontal portion BC, repre-
sents the vapor pressures, at 88°.5 C., of mixtures of isobutyl
alcohol and water where two layers are present. The vapor

SOLUTIONS 181

pressure of the homogeneous mixtures are represented by AB and
CD, AB corresponding to solutions of isobutyl alcohol in water,
and CD to solutions of water in isobutyl alcohol. The dotted
line A'B'C'D' represents the boiling-points of all possible mix-
tures of isobutyl alcohol and water, under normal atmospheric
pressure.

Solutions of Solids in Liquids. The solubility of a solid in a
liquid is limited and is dependent upon the temperature, the
nature of the solute and the nature of the solvent. When a
solvent has taken up as much of a solute as it is capable of dis-
solving at a definite temperature, the solution is said to be satu-
rated. There are two general methods for the preparation of
saturated solutions: — (1) An excess of the finely-divided solute
is agitated with a known amount of the solvent, at a definite
temperature, until equilibrium is attained; (2) the solvent is
heated with an excess of the solute to a temperature higher than
that at which saturation is required, and then cooled in contact
with the solid solute to the desired temperature. Both of these
methods give equally satisfactory results provided sufficient
time is allowed for the establishment of equilibrium, and provided
the solid substance is always present in excess. The solubility
of a solid in a liquid may be expressed as the number of grams of
the solute in a given mass or volume of solvent or solution, but
it is usually expressed as the number of grams of solute in 100
grams of solution. The solubility of solids has recently been
shown to be somewhat dependent upon their state of division.
Thus, Hulett * has found that a saturated solution of gypsum at
25° C. contains 2.080 grams of CaSO4 per liter, whereas when
very finely divided gypsum is shaken with this solution, it is
possible to increase the content of dissolved CaSO4 to 2.542 grams
per liter. When a saturated solution is cooled, every _ trace of
solid solMeJbSi^^ dissolved solid may not

separate. jSuchLa solution is said" ^^bWsupef saturated.

As ITgeneral r^^tTG^sduEniEy 15F's5li^ TiTTTq^ids increases
with the temperature, as shown in Fig. 57. Several exceptions
to this rule are known, among which may be mentioned calcium

* Jour. Am. Chera. Soc., 27, 49 (1905).

182

THEORETICAL CHEMISTRY

hydroxide, calcium sulphate above 40° C., and sodium sulphate
between the temperatures of 33° C. and 100° C.

Solubility curves are usually continuous, but exceptions to this
rule are common: the solubility curve of sodium sulphate fur-

40 00 80 100

Temperature

Fig. 57.

nishes an illustration. The discontinuity in the solubility curve
of sodium sulphate is due to the fact that we are not dealing with
one solubility curve, but with two solubility curves. At temper-
atures below 33° C., the dissolved salt is in equilibrium with the
decahydrate, Na-jSO^lOH^O, whereas at temperatures above
33° C. the dissolved salt is in equilibrium with the anhydrous
salt, Na2S04. The solubility of Na2S04.10H2O increases with
the temperature, while the solubility of Na^SC* diminishes. That
we are actually dealing with two solubility curves, is proved by
the fact that the solubility curves of the hydrated and anhydrous
salts in supersaturated solutions are continuations of the corre-
sponding curves for saturated solutions, as shown by the dotted

SOLUTIONS 183

curves in Fig. 57. If we select any point, such as p, lying between
a dotted curve and a full curve, it is apparent that it represents a
solution supersaturated with respect to Na^SO^lOH^O, but un-
saturated with respect to Na2S04. If pure anhydrous sodium
sulphate be shaken with this solution it will slowly dissolve, where-
as if a trace of the hydrated salt be added, the solution will deposit
Na2SO4.10 H20, until the amount remaining in solution corre-
sponds to the solubility of the hydrate at that temperature.
Supersaturated solutions of some substances can be preserved
indefinitely, provided all traces of the solid phase are excluded.
Such solutions are called metastable. On the other hand there
are some supersaturated solutions which deposit the excess of
solid solute even when all traces of it are excluded. These solu-
tions are termed labile. The distinction between metastable and
labile solutions is not sharp. If a metastable solution is suffi-
ciently cooled, or if its concentration is sufficiently increased, it
may be made to pass over into the labile condition. The concen-
tration at which this tradition occurs is termed the metastable
limit. The stability of supersaturated solutions has recently
been shown by Young * to be greatly influenced by vibrations or
sudden shocks within the solution. He has been able to control
the amount of overcooling in a supersaturated solution, by alter-
ing the intensity of the vibrations due to the friction between glass
or metal surfaces within the solution.

Very little is known concerning the relation between solubility
and the specific properties of solute and solvent.

Owing to the fact that the change in volume resulting .from the
solution of a solid in a liquid is very small, the effect of pressure
on the solution is almost negligible. The chief factors condition-
ing the change in solubility due to increasing pressure, are the
heat of solution of the solute in the nearly saturated solution,
and the change in volume on solidification. Very few experiments
have been made to determine the effect of pressure on solubility,
van't Hoff states that the solubility of a solution of ammonium
chloride, a salt which expands when dissolved, decreases by 1 per
cent for 160 atmospheres, while the solubility of copper sulphate,
* Jour. Am. Chem. Soc., 33, 148 (1911).

184

THEORETICAL CHEMISTRY

a salt which contracts when dissolved, increases by 3.2 per cent
for 60 atmospheres.

Solid Solutions. In general, when a dilute solution is suffi-
ciently cooled the solvent separates in the form of crystals which
are almost entirely free from the solute. When, however, the
temperature of a solution of iodine in benzene is reduced to the
freezing-point, the crystals which separate are found to contain
iodine^ Furthermore, the depression of the freezing-point of the
solvent is found to be less than that calculated on the assumption
that the solvent crystallizes uncontaminated with the solute.
Such solutions were first studied by van't Hoff.* He found that
when the concentration of such abnormal solutions is varied, the
ratio of the amount of solute in the liquid solvent to the amount
of solute in the solidified sol vent* remains constant. Thus, in solu-
tions of iodine in benzene, the ratio of the concentration of iodine
in the liquid to its concentration in the crystallized benzene is
constant. In the following table Ci is the concentration of iodine
in the liquid benzene, and fy is the concentration of iodine in the
solid benzene.

Cj/C2

3 39
2 587
0.945

1 279
0.925
0.317

0 377
0 358
0 336

van't Hoff pointed out the analogy between the distribution of
the solute between the solid and liquid solvent, and the distribution
of a gas between a liquid and the free space above it. In other
words, the distribution follows Henry's law for the solution of a
gas in a liquid. Since the crystals containing both solute and
solvent are perfectly homogeneous, van't Hoff suggested that
they be regarded as solid solutions. The mixed crystals which
separate from solutions of isomorphous substances being chem-
ically and physically homogeneous, are to be considered as solid
solutions. Many alloys possess the properties characteristic of

* Zeit. phys. Chem., 5, 322 (1890).

SOLUTIONS 185

solid solutions; hardened steel, for example, being regarded as a
homogeneous solid solution of carbon in iron. One of the char-
acteristic properties of a dissolved substance is its tendency to
diffuse into the pure solvent. Interesting experiments performed
by Roberts- Austen * have shown that even solids have the prop-
erty of mixing by diffusion. Thus, by keeping gold and lead in
contact at constant temperature for four years, he was able to
detect the presence of gold in the layer of lead at a distance of
7 mm. from the surface of separation. Many other instances of
diffusion in solids have been observed.f

Instances of gases and liquids dissolving in solids are also
known. Thus platinum, palladium, charcoal and other sub-
stances have the property of taking up large volumes of hydrogen.
This phenomenon, known as occlusion, is but little understood,
van't Hoff has suggested that wllen~1iydrogen dissolves in palla-
dium we are really dealing with two solid solutions: one a solu-
tion of hydrogen in palladium and the other a solution of
palladium in solid hydrogen, the system being analogous to
that of two partially miscible liquids.

Certain natural silicates, the so-called jeolites,_are transparent
and homogeneous. Since they contain varying quantities of
water they may be regarded as examples of solutions of liquids
in solids. This classification is further justified by the fact that
portions of the water may be removed and replaced by other
substances, such as alcohol, with apparently no change in the
transparency or homogeneity of the mineral.

PROBLEMS.

1. 2.3 liters of hydrogen under a pressure of 78 cm. of mercury, and
5.4 liters of nitrogen at a pressure of 46 cm. were introduced into a ve&sel
containing 3.8 liters of carbon dioxide under a pressure of 27 cm. What
was the pressure of the mixture? Ans. 140 cm. of mercury.

2. Air is composed of 20.9 volumes of oxygen and 79.1 volumes of
nitrogen. At 15° C. water absorbs 0.0299 volumes of oxygen and 0.0148

* Proc. Roy. Soc., 67, 101 (1900).

t See Report on Diffusion in Solids, by C. H. Desch, Chem. News, 106,
153 (1912).

186 THEORETICAL CHEMISTRY

volumes of nitrogen, the pressure of each being that of the atmosphere.
Calculate the composition of the mixture of gases absorbed by the water.
Ans. 34.8% by vol. of 0, and 65.2% by vol. of N.

3. The vapor pressure of the immiscible liquid system, aniline-water,
is 760 mm. at 98° C. The vapor pressure of water at that temperature is
707 mm. What fraction of the total weight of the distillate is aniline.

Ans. 0.28.

4. The boiling-point of the immiscible liquid system, naphthalene-water,
is 98° C. under a pressure of 733 mm. The vapor pressure of water at
98° C. is 707 mm. Calculate the proportion of naphthalene in the dis-
tillate. Ans. 0.207.

CHAPTER X.
DILUTE SOLUTIONS AND OSMOTIC PRESSURE.

Osmotic Pressure. In the preceding chapter reference was
made to the fact that diffusion is a characteristic property of solu-
tions. If a few cubic centimeters of a concentrated solution of
cane sugar are placed at the bottom of a tall cylinder, and water
is added, care being taken to prevent mixture, the sugar immedi-
ately begins to diffuse into the water, the process continuing until
the concentration of sugar is the same throughout the liquid.
The sugar molecules move from a region of high concentration
to a region of low concentration, the rate of diffusion being rela-
tively slow owing to the viscosity of the medium. A similar
process is encountered in the study of gases, but the rate of gas-
eous diffusion is extremely rapid. In terms of the kinetic theory,
the movement of the molecules of a gas from regions of high
concentration to regions of low concentration, is to be considered
as due to the pressure of the gas. By analogy, we may regard the
process of diffusion in solutions as a manifestation of a driving
force, known as the osmotic pressure.

Semi-permeable Membranes. The use of a semi-permeable
membrane for the measurement of the partial pressure of nitrogen
in a mixture of nitrogen and hydrogen, has already been explained.
A similar method may be employed for the measurement of
the osmotic pressure of a solution, provided a suitable semi-perme-
able membrane can be found. Such a membrane must prevent
the passage of the molecules of solute and must be readily perme-
able to the molecules of solvent; it must exert a selective action
on solute and solvent. If a solution is separated from the pure
solvent by a semi-permeable membrane, diffusion of the solute
is no longer possible. Since equilibrium of the system can only
be attained when the concentrations on both sides of the mem-
brane are equal, it follows that the solvent mult pass through

187

188 THEORETICAL CHEMISTRY

the membrane and dilute the more concentrated solution. A
number of semi-permeable membranes have been discovered
which are readily permeable to water and nearly, if not entirely,
impermeable to various solutes. About the middle of the eight-
eenth century Abb6 Nollet discovered that certain animal mem-
branes are permeable to water but not to alcohol.

Artificial semi-permeable membranes were first prepared by
M. Traube.* If a glass tube, provided with a rubber tube and
pinch-cock, be partially filled, by suction, with a solution of
and then immersed in a solution of potassium

ferrocyanide, a thin film of copper ferrocyanide_ will be formed
at the junction of the two solutions. When the film has once
been formed, further precipitation of copper ferrocyanide will
cease, the solutions on either side of the film remaining clear.
Traube showed that this membrane is semi-permeable. He also
showed that a number of other gelatinous precipitates possess
the property of semi-permeability. A membrane formed in the
above manner is easily ruptured and is wholly inadequate for
quantitative or even qualitative experiments. Pfeffer f devised
a method for strengthening the membrane. By depositing the
precipitate in the walls of a porous clay cup, the area of unsup-
ported membrane is greatly diminished and its resisting power
correspondingly increased. Pfeffer directs that the cup to be
used for this purpose must be thoroughly washed, and its walls
allowed to become completely permeated with water. The cup
is then filled to the top with a solution of copper sulphate, con-
taining 2.5 grams per liter, and allowed to stand for several hours
in a solution of potassium ferrocyanide, containing 2.1 grams
per liter. The two solutions diffuse through the walls of the cup
and on meeting, deposit a thin membrane of copper ferrocyanide.
When precipitation is complete, the cup is thoroughly washed
and soaked in water. The cup is then filled to the top with a
solution of cane sugar, and a rubber stopper, fitted with a long
glass tube of narrow bore, is inserted, care being taken to exclude
air-bubbles. The stopper is then made fast with a suitable

* Archiv. fur Anat. und Physiol., p. 87 (1867).
t demotische Untersuchungen, Leipzig, 1877.

DILUTE SOLUTIONS AND OSMOTIC PRESSURE

189

cement, and the cup completely immersed in a beaker of water.
The completed apparatus is shown in Fig. 58. If the formation
of the membrane has been successful, the level of the liquid in the
vertical glass tube will slowly rise and will eventually attain a
height of several meters. If the mem-
brane is sufficiently strong and no leaks
develop, the passage of water through the
membrane will continue until the hydro-
static pressure of the column of liquid in
the tube is great enough to overcome the
tendency of the water to force its way
into the sugar solution. As a general
rule, the membrane becomes ruptured be-
fore equilibrium is attained.

Measurement of Osmotic Pressure.
The first direct measurements of osmotic
pressure were made by Pfeffer. His ex-
periments deserve brief consideration,
since the results obtained furnish the
basis of the modern theory of solution.
The cell used was similar to that described
above, but instead of employing a ver-
tical glass tube as a manometer, the cup
was connected, as shown in Fig. 59, with
a closed mercury manometer. The sub-
stitution of the closed for the open man-
ometer is necessitated by the fact, that
with an open manometer so much water entered the cell that
the concentration of the solution became appreciably diminished,
and the pressure actually measured corresponded to a solu-
tion of smaller concentration than that introduced into the cell.
With the closed manometer, when a trace of water has entered
the cell, sufficient pressure is developed to prevent the further en-
trance of more water. Pfeffer calculated that with a cell, the
capacity of which was 16 cc., the volume of water entering before
equilibrium was attained, did not exceed 0.14 cc. In his experi-
ments, Pfeffer determined the density of the cell Contents before

Fig. 58.

190

THEORETICAL CHEMISTRY

and after measurement of the osmotic pressure, and corrected for
any change in concentration. With this apparatus he made numer-
ous measurements of the osmotic
pressures of different solutions, the
entire apparatus being immersed in
a constant-temperature bath. With
solutions of cane sugar he obtained
the results given in the accompany-
ing table, where C denotes the per-
centage concentration of the solu-
tion, and P the corresponding
osmotic pressure, expressed in centi-
meters of mercury. The tempera-
ture varied from 13.5° C. to 14.7° C.

P/C

53.5

101.6

50.8

208.2

52.0

307.5

51.2

It is evident from these results,
that the osmotic pressure is propor-
tional to the concentration of the
solution, since P/C is approximately
constant. The deviations from con-
stancy in the ratio of pressure to
concentration may be ascribed to
experimental errors, since the dif-
ficulties involved in these measure-

^ ments are very great. Pfeffer also

Fig. 59. studied the influence of temperature

on osmotic pressure, and showed

that as the temperature is raised the pressure increases. The
following table gives his results for a 1 per cent solution of cane
sugar.

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 191

Temperature.

Osmotic Pressure.

6° 8
13°. 2
14°. 2
22° 0
36°. 0

cin.

50.5
52 1
53.1
54 8
56.7

Osmotic Pressure and the Nature of the Membrane. Pfeffer
also studied the effect of the nature of the membrane on osmotic
pressure. In addition to copper ferrocyanide, he used membranes
of calcium phosphate and Prussian blue. His results seemed to
indicate that the magnitude of the osmotic pressure developed,/
was dependent upon the nature of the membrane used.1

The variations observed have since been shown to have been
due to leakage of the calcium ^phosphate and Prussian blue mem-
branes, the copper ferrocyanide membrane Being the only one
which was capable of withstanding the pressure. Ogfagfrld *. has
devisedjan ingepj^&Jheoretical demonstration of the Jact
osm^c^pressurejnust^be independent of t&e natm^of_the
brane employedjn^n^simng it r" Let A and B, in Fig. 60, repre-

Fig. 60.

sent two different semi-permeable membranes placed in a glass
tube of wide bore. Let us imagine the space between the two
membranes to be filled with a solution, and the tube immersed
in a vessel of water. If the osmotic pressures developed at A
and B are pi and p* respectively, and p2 is less than pi, then
water will pass through A until the pressure pi is reached. Since
the pressure at B only reaches the value p2, however, the pres-
sure pi can never be attained, and a steady stream of water from
A to By under the pressure pi — p2, will result. This, however,
would be a perpetual motion, and since this is impossible, the
osmotic pressures at the two membranes must be the same.
* Lehrb. d. allg. Chem., I., p. 662.

192

THEORETICAL CHEMISTRY

Theoretical Value of Osmotic Pressure. The physico-chem-
ical significance of Pfeffer's results was first perceived by van't
Hoff.* In a remarkably brilliant paper, he pointed out the
existence of a striking parallelism between the properties of gases
and the properties of dissolved substances.

We have already called attention to the analogy between osmotic
pressure and gas pressure: we now proceed to trace the connec-
tion between osmotic pressure, volume and temperature, as first
pointed out by van't Hoff. Pfeffer's experiments showed that at
constant temperature, the ratio, P/(7, is constant for any one
solute. Since the concentration varies inversely as the volume
in which a definite amount of solute is dissolved, we obtain, by
substituting l/V for C, the equation, PV = constant, which is
plainly the analogue of the familiar equation of Boyle for gases.
An examination of Pfeffer's data for osmotic pressures at differ-
ent temperatures, convinced van't Hoff that the law of Gay-
Lussac is also applicable to solutions.

In the following table, the osmotic pressures in atmospheres for
a 1 per cent solution of cane sugar at different temperatures are
recorded, together with the pressures calculated on the assump-
tion that the osmotic pressure is directly proportional to the
absolute temperature.

Temperature.

/' (oba.)

P (calc ).

6° 8

0 664

0.665

13° 7

0 691

0 681

15°. 5

0 684

0 686

22° 0

0 721

0.701

32° 0

0.716

0 725

36° 0

0.746

0.735

Since the laws of Boyle and Gay-Lussac are both applicable,
we may write an equation for dilute solutions corresponding to
that already derived for gases, or

PV = B'T.

in which P is the osmotic pressure of a solution containing a defi-
* Zeit. phys, Chem., i, 481 (1887).

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 193

nite weight of solute in the volume, V, of solution, T being the
absolute temperature of the solution and Rl a constant corre-
sponding to the molecular gas constant.

The molecular gas constant R has already been evaluated and
has been found to be equal to 0.0821 liter-atmosphere.

Making use of Pfeffer's data, van't Hoff calculated the value
of Rf in the above equation, in the following manner: the osmotic
pressure of a 1 per cent solution of cane sugar at 0° C. is 0.649
atmosphere, and since the concentration of the solution is 1 per
cent, the volume of solution containing 1 mol of sugar, will be
34,200 cc. or 34.2 liters. Substituting these values in the equa-
tion, we have

n, PV 0.649X34.2 AACM01.,

R' = -=- = - ;r-- - = 0.0813 hter-atmos.,

a value which is nearly the same as that of the molecular gas
constant, R. The equality of R and Rf leads to a conclusion of
the greatest importance, as was pointed out by van't Hoff, viz.,
"the osmo&(L%ressure exerted by any substance in solution is the same
as it u)ouldi exj^t if present as a gas in the same volume as that occupied
by the solution, provideSTihat the solution is so dilute that the volume

•tSrojisMtaWMiMcffTOR'. »v> w ' '"**•* ' *

occupied by the solute is negligible in comparison with that occupied
by the solvent" It should be remembered that we are not justified
in concluding from this proposition of van't Hoff, that osmotic
pressure and gaseous pressure have a common origin. While
the origin of osmotic pressure may be kinetic, it is also conceivable
that it may result from the mutual attraction of solvent and
solute, or that it may bear some relation to the surface ten-
sion of the solution. Up to the present time no wholly satis-
factory explanation of the cause of osmotic pressure has been
advanced.

Just as 1 mol of gas at 0° C. and 760 mm. pressure occupies a
volume of 22.4 liters, so when 1 mol of a substance is dissolved
and the solution diluted to 22.4 liters at 0° C., it will exert an
osmotic pressure of 1 atmosphere. I^therj^ords, woZor weights,
or quantities proportional to molar weights, of different substances,
wKen diss^KTirTeqiu^ volumes of the same solvent exert the same

194 THEORETICAL CHEMISTRY

osmotic pressure. If we deal with n mols of solute instead of 1 mol
the general equation becomes

PV = nRT.

But n = gf/M, where g is the number of grams of solute per liter,
and M is its molecular weight. Substituting in the preceding
equation, we have

or,

M - gRT-
M~~

Since P, V, g, R and T are all known, M can be calculated. The
direct measurement of the osmotic pressure of a solution does not
afford a practical method for the determination of the molecular
weight of dissolved substances, because of the experimental
difficulties involved and the time required for the establishment
of equilibrium. There are other and simpler methods for deter-
mining molecular weights in solution, based upon certain proper-
ties of solutions which are proportional to their respective osmotic
pressures.

Recent Work on the Direct Measurement of Osmotic Pressure.
It is only within the past decade that the investigations of Pfeffer
have been confirmed and extended by elaborate and systematic
experiments on the direct measurement of osmotic pressure.
Morse and his co-workers,* while employing a method essentially
the same as that of Pfeffer, have, as the result of much patient
labor, brought the apparatus to such a high state of perfection,
that the experimental errors are now estimated to affect only
the second place of decimals in the numerical data expressing
osmotic pressures in atmospheres. The most important of the
improvements introduced by Morse are the following: — (1) the
improvement of the quality of the membrane; (2) the improve-
ment of the connection between the cell and the manometer,
and (3) the improvement of the means of accurately measuring

* Am. Chem. Jour., 26, 80 (1901); 34, 1 (1905); 3$, 1, 39 (1906); 37, 324,
425, 558 (1907); 381 175 (1907); 39, 667 (1908); 40, 1, 194 (1908); '41, 1, 257
(1909).

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 195

the pressure. The membrane of copper
ferrocyaiiide is deposited electrolyti-
cally. After thorough washing and
soaking in water, the porous cup, made
from specially prepared clay, is filled
with a solution of potassium ferrocy-
anide and immersed in a solution of cop-
per sulphate. An electric current is
then passed from a copper electrode
in the solution of copper sulphate, to
a platinum electrode immersed in the
solution of potassium ferrocyanide.
This drives the copper and ferrocy-
anide ions toward each other, and the
membrane of copper ferrocyanide is
thus formed in the walls of the cup.
The passage of the current is continued
until the electrical resistance reaches a
value of about 100,000 ohms. The
cell is then rinsed, and soaked in water
for several hours, and then the electro-
lytic treatment is repeated until the
electrical resistance attains a maximum
value. A solution of cane sugar is now
introduced into the cell, which is con-
nected with the manometer and im-
mersed in water. When the pressure
has attained its maximum value, the
apparatus is dismantled and the cell,
after thorough washing and soaking in
water, is again subjected to the electro-
lytic process of membrane forming. In
this way the weak places in the mem-
brane which may have yielded to the
high pressure, can be repaired, and by
continued repetition of this treatment
the membrane can ultimately be brought to its maximum power

Fig. 61.

196

THEORETICAL CHEMISTRY

of resistance. A sketch of the Morse apparatus is shown in Fig.
61. A description of the details of this apparatus lies beyond the
scope of this book. The results of the work of Morse and his
students are of the highest importance. The osmotic pressures
of solutions of cane sugar and dextrose have been shown to be
proportional to the respective concentrations, provided the con-
centration is referred to unit volume of solvent instead of unit
volume of solution. Thus in their experiments, the solutions were
made up containing from 0.1 to 1.0 mol of solute in 1000 grams of
water. Morse calls such solutions weight-normal solutions in con-
trast to volume-normal solutions, in which 1 rnol or a fraction of
a mol of solute is dissolvpd in water and the solution diluted to 1
liter. The following data taken from the work of Morse, shows
that when concentration is expressed on the weight-normal basis,
there is direct proportionality between osmotic pressure and con-
centration. The figures refer to solutions of dextrose at 10° C.

Molar Concentration.

Osmotic Pressure.

Per 1000 gm.
Water.

Per Liter of
Solution.

In Atinos.

Relative to First
as Unity.

0.1
0 2
0 5
1.0

0 099
0 196
0 474
0 901

2 39
4 76
11 91
23.80

1.00
1.99
4.98
9.96

Morse and his co-workers also conclude from their experiments
at temperatures ranging from 0° C. to 25° C., that^hg^eigp^r-
ature coefficients of osmotic pressure and gas pressure are prac-
ticaUx<MJdentical. In other words, their results confirm the
conclusions of van't Hoff , that the law of Gay-Lussac is applicable
to solutions. The results of the experiments of Morse are of
special interest in connection with the proposition of van't Hoff,
that the osmotic pressure of a dilute solution is the same as that
which the solute would exert if it were gasified at the same temper-
ature and occupied the same volume as the solution. The data
in the following table is taken from the work of Morse on solutions

DILUTE SOLUTIONS AND OSMOTIC PRESSURE

197

of cane sugar at 15° C. In addition to the observed osmotic
pressures, the table contains the corresponding gas pressures,
calculated (1) on the assumption that the solute when gasified
occupies the same volume as the solution (proposition of van't
Hoff), and (2) on the assumption that it occupies the same volume
as the solvent alone.

Molar Concentration.

Osmotic Pressure in Atmos.

Per 1000 gm. of
Water.

Per Liter of
Solution.

Obs.

Calc. (a).

Calc. Cb).

0.1

0 098

2 48

2 30

2 35

0 2

0 192

4 91

4 51

4 70

0 4

0 369

9 78

8 67

9 40

0 6

0 533

14 86

12 51

14 08

0 8

0 684

20 07

16 07

18 79

1 0

0 825

25 40

19.38

23.49

The calculated pressures, recorded in the last column, are in
much closer agreement with the observed osmotic pressures, than
are the calculated pressures, recorded in the fourth column of the
table. The proposition of van't Hoff should then be modified
to read as follows: — A dissolved substance in dilute solution^^x^rts
an osmotic pressure equal to that which it wpuld exert if it were gas-
ified at the same temperature, and the volume of the gas were reduced
to that of the solvent in the pure state., The investigations of Morse
and his co-workers may be summarized thus: — (1) the law of
Boyle is applicable to dilute solutions, provided the concentration
is referred to 1000 grams of solvent and not to 1 liter of solution;
(2) the law of Gay-Lussac is also applicable to dilute solutions,
that is, the temperature coefficients of osmotic pressure and gas
pressure are equal, and (3) the small departures from the theo-
retical values of the osmotic pressures may be traced to hydration
of the solutec

Direct measurements of the osmotic pressure of concentrated
solutions of cane sugar, dextrose and mannite have been made by
the Earl of Berkeley and E. G. J. Hartley.* The method em-

* Proc. Roy. Soc., 73, 436 (1904); Trans. Roy. Soc. A., 206, 481 (1906).

198 THEORETICAL CHEMISTRY

ployed by these investigators is slightly different from that of
Pfeffer or Morse; the tendency of water to pass through the
semi-permeable membrane is offset by the application of a counter
pressure to the solution. A membrane of copper ferrocyanide
is deposited electrolytically very near the outer surface of a tube
of porous porcelain. This tube is placed co-axially within a large
cylindrical vessel of gun metal, an absolutely tight joint between
the two being secured by an ingenious system of dermatine rings
and clamps. The open ends of the porcelain tube are closed bv
rubber stoppers fitted with capillary tubes bent at right angles)
one of the latter being provided with a glass stop-cock. When a
determination of osmotic pressure is to be made, the apparatus is
placed in a horizontal position and water is introduced into the
porcelain tube, completely filling it and the connecting capillary
tubes up to a certain level. The gun metal vessel is then filled
with the solution, and connected with an auxiliary apparatus by
means of which a gradually increasing hydrostatic pressure can
be applied. If no pressure is applied to the solution, water will
pass through the semi-permeable membrane into the solution, and
the level of the water in the capillary tubes will fall. In carrying
out a measurement, therefore, as soon as the solution is introduced
into the gun metal vessel, hydrostatic pressure is applied, the mag-
nitude of the pressure being so adjusted as to counterbalance the
osmotic pressure of the solution. The level of the water in the
capillary tubes serves to indicate the relative magnitudes of
the osmotic and hydrostatic pressures. When the level of the
water in the capillary tubes remains constant, the two pressures
are in equilibrium. The following are the values of the equilibrium
pressures of solutions of cane sugar, dextrose and mannite at 0° C.
It must be remembered that when the two pressures are in equi-
librium, there is always a pressure of one atmosphere on the solvent.
As will be seen the pressures developed in the more concen-
trated solutions are enormous and it is a surprising fact, that even
in cases where the highest pressures were measured, hardly a
trace of sugar was found in the pure solvent, the membrane
retaining its property of semi-permeability throughout the entire
range of pressures* The figures in the third column are calculated

DILUTE SOLUTIONS AND OSMOTIC PRESSURE

CANE SUGAR.

Osmotic Pressure in Atmospheres.

Cone. gm. per

Liter.

Obs.

Calc.

180.1

13 95

13.95

300.2

26.77

28 74

420.3

43.97

32.55

540.4

67.51

41 85

660.5

100 78

51 16

750.6

133 74

58.14

DEXTROSE.

* ^~* *

Osmotic Pressure in Atmosphere.

Cone. gm. per

V-

Liter.

Obs.

Calc.

99 8

13 21

199.5

29.17

26.41

319 2

53.19

42.25

448 6

87.87

59 28

548 6

121 . 18

72 61

MANNITE.

Cone, gm, per
Liter.

Osmotic Pressure in Atmospheres.

Obs.

Calc.

100
110
125

13 1
14 6
16.7

13.1
14.4
16.4

on the assumption that there is direct proportionality between
osmotic pressure and concentration. It is apparent that in every
case the observed osmotic pressure is greater than the calculated.
Even when the concentrations are expressed on the weight-normal

200

THEORETICAL CHEMISTRY

basis, as recommended by Morse, the osmotic pressure increases
more rapidly than the concentration.

This is well shown in the accompanying diagram, Fig. 62, due
to the Earl of Berkeley. In this diagram, the osmotic pressures
of solutions of cane sugar are plotted against concentrations, curve
A representing the actually observed osmotic pressures; curve C

120

100

380

E60

§40

100 200 300 400 600

Grams Cfetne Sugar per liter of Solution

Fig. 62.

600

700

being traced on tlie assumption that osmotic pressure may be
calculated from the equation, PV = JK77, where V denotes the
volume of solvent containing 1 mol. of cane sugar; and curve B,
a straight line, being drawn on the assumption that osmotic
pressure may be calculated from the equation, PV = RT, where
V represents the volume of solution containing 1 mol.
While the theoretical and observed values of the osmotic pres-

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 201

sure are approximately equal in the more dilute solutions, it is
obvious that the observed values of the osmotic pressure of the
concentrated solutions are always greater than the calculated
values, even when the calculation is made on the assumption that
V in the equation, PV = RT, is the volume of the solvent. The
abnormally high osmotic pressures observed by the Earl of
Berkeley have been discussed by Callendar * who suggests hydra-
tion of the solute as a probable cause.

He shows, that if 5 molecules of water are assumed to be asso-
ciated with each molecule of cane sugar in the most concentrated
solutions studied by the Earl of Berkeley, the discrepancy between
the observed and calculated values of the osmotic pressure dis-
appears.

Comparison of Osmotic Pressures. Although the difficulties
involved in the direct determination of osmotic pressure are
many, these can be avoided by the employment of one of several
indirect methods which have been devised for the comparison of
osmotic pressures. All of these methods depend upon the exchange
of water which occurs when two solutions are separated by a
semi-permeable membrane. The movement of the water will
always be in such a direction as to tend to equalize the osmotic
pressures on opposite sides of the membrane, or, in other words,
.the transfer of water will take place from the solution with the
lesser osmotic pressure to the solution with the greater osmotic
pressure.

The Plasmolytic Method. In this method, solutions of various
substances are prepared, the concentration of each being such
that its osmotic pressure is the same as that of a particular plant
cell. Obviously the osmotic pressures of all of these solutions
must be equal: such solutions are said $P^i&li3ML£* ?'s°sm?*iJL
The plasmolytic method for the comparison of osmotic ^pressures
was developed by the Dutch botanist, De Vries.f This method
depends upon the shrinking or swelling of the protoplasmic sac
of plant cells when they are immersed in a solution whose osmotic

* Proc. Roy. Soc. A., 80, 466 (1908).

t Jahrb.wiss.Botanik., 14,427(1884); Zeit. phys. Chem., 2, 415 (1888); 3,
103 (1889).

202 THEORETICAL CHEMISTRY

pressure differs from that of their own sap. De Vries found that
the cells of Tradescantia discolor. Curcuma rubricaulis, and Begonia
manicata fulfil the necessary conditions, viz.; the cell walls are
strong and resist alteration when immersed in solutions, the cells
are readily permeable to water, and the cell contents are colored,
thus enabling the slightest contraction or expansion to be de-
tected. The cell walls are lined on the inside with a thin, elastic,
semi-permeable membrane which encloses the colored contents
of the cell. The content of the cell consists of an aqueous solu-
tion of several substances, among which may be mentioned
glucose, potassium and calcium malate, together with coloring
matter. The osmotic pressure of the cell contents ranges from
four to six atmospheres. The semi-permeable membrane expands
when the contents of the cell increases and contracts when the
contents diminishes. In making a comparison of osmotic pres-
sures by this method, tangential sections are cut from the under
side of the mid-rib of the leaf of one of the above plants, e.g.,
Tradescantia discolor, and are placed in the solution whose osmotic
pressure it is desired to compare with that of the cell contents.
The cells are then observed under the microscope, any decrease
in pressure below the normal resulting in a detachment of the
semi-permeable membrane from one or more points of the cell
wall. This contraction always occurs when the cells are im-
mersed in a concentrated solution, the phenomenon being termed
plasmolysis. When the solution in which the cells are placed
has a lower osmotic pressure than the cell contents, no visible
effect is produced, the increased pressure within the cell simply
forcing the membrane closer to the rigid cell walls. By starting
with a concentrated solution, the osmotic pressure of which is
greater than that of the cell, and gradually diluting it, a concen-
tration will ultimately be reached at which the elastic membrane
will just completely fill the cell. This solution is isotonic with
the cell contents. In this method the very reasonable assump-
tion is made that all of the cells have the same osmotic pressure,
any differences which might have existed having equalized them-
selves in the living plant. The microscopic appearance of cells
of Tradescantia discolor when immersed in different solutions is

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 203

shown in Fig. 63. The appearance of the normal cell when im-
mersed in water or in a solution whose osmotic pressure is less
than that of the cell contents, is shown in A. When the cell is
immersed in a 0.22 molar solution of cane sugar it appears as in
JB, this solution having a greater osmotic pressure than the cell
contents. When the cell is immersed in a molar solution of

Fig. 63.

potassium nitrate, there is marked plasmolysis, as shown in C.
De Vries determined the concentrations of a large number of
solutions which were isotonic with the cell contents. He ex-
pressed his results in terms of the isotonic coefficient, which he
defined as the reciprocal of the molar concentration. The iso-
tonic coefficient of potassium nitrate was taken equal to 3. A
few of De Vries' results are given in the following table.

Substance.

Formula.

Isotonic
Coefficient.

Glycerol

C3H8O3

1 78

Glucose

CeH^Oe

1 81

Cane sugar

\*t 1 2-ti 22 *J 1 1

1 88

Malic acid

C4HeO5

1 98

Tartaric acid

CiHUOg

2 02

Citric acid

CeHgO?

2 02

Potassium nitrate

KNO8

3 00

Magnesium chloride

MgCl2

4 33

204 THEORETICAL CHEMISTRY

De Vries applied the plasmolytic method to the determination
of the molecular weight of raffinose. At that time there was
considerable uncertainty as to the correct formula of crystallized
raffinose, three different formulas, all consistent with the results
of analysis, having been proposed as follows: — Ci8H32Oi6.
5 H20, Ci2H22Oii.3 H20, and C36H64032. De Vries found that a
3.42 per cent solution of cane sugar was isotonic with a 5.96 per-
cent solution of raffinose. Letting the unknown molecular
weight of raffinose be represented by M9 then

3.42 : 5.96 :: 342 : M.

Solving the proportion we have, M - 596. This result has since
been confirmed by purely chemical methods, and the formula,
CisH.tfOie.5 H2O, the molecular weight of which is 594, is thus
established.

The Blood Corpuscle Method. The red blood corpuscle is a
cell, the contents of which is enclosed by a thin elastic semi-
permeable membrane. Unlike the plant cells, there is no resistant
cell wall to give support to the membrane, so that when red blood
corpuscles are immersed in water they at first swell, owing to the
osmotic pressure developed, and finally burst. When the mem-
brane is ruptured, the coloring matter of the cell, the haemoglobin,
escapes and the water acquires a deep red color.

Advantage of this behavior of red blood corpuscles was taken
by Hamburger * for the comparison of osmotic pressures. He
found that when a 1.04 per cent solution of potassium nitrate
is added to the defibrinated blood of a bullock, the corpuscles
will settle completely to the bottom, while the supernatant liquid
will remain clear. On the other hand, if a 0.96 per cent solution
of potassium nitrate is used, the corpuscles will not settle and the
supernatant liquid becomes colored. If more dilute solutions of
potassium nitrate are used, the solution acquires a still deeper
color. By careful adjustment, a concentration of potassium
nitrate can be found in which the red blood corpuscles will just
settle. In like manner, the concentration of solutions of other
substances can be so adjusted as to cause the precipitation of the

* Zeit. phys. Chem., 6, 319 (1890).

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 205

corpuscles. These solutions are isotonic. Without going into
details, it may be said that the isotonic coefficients obtained by
Hamburger, agree well with those obtained by the plasmolytic
method,

The Hffimatocrit Method* In this method developed by
Hedin,* advantage is again taken of the properties of red blood
corpuscles. As has already been stated, when red blood cor-
puscles are immersed in solutions of gradually diminishing
concentration of the same solute, they continue to swell and ulti-
mately the semi-permeable envelope bursts. On the other hand,
when the corpuscles are immersed in solutions of gradually increas-
ing concentration, they shrink, owing to the transfer of water
from the corpuscles. It is apparent that there must be a certain
concentration for each solute which will cause no change in the
volume of the corpuscles. To determine this concentration, use
is made of an instrument known as an hcematocrit. This is
simply a graduated thermometer-tube which may be attached to
the spindle of a centrifugal machine. When the spindle is re-
volved at high speed, the corpuscles collect in the bottom of the
graduated tube. A measured volume of blood is centrifuged
until no further shrinkage in volume of the corpuscles can be
detected in the haematocrit. The same volume of blood is then
added to each of a series of solutions whose concentration dimin-
ishes progressively, and the volume of the corpuscles is determined
as in pure blood. In this way the concentration of the solution
is found, in which the volume of the corpuscles is the same as in
the undiluted blood. By proceeding in a similar manner with
solutions of different substances, a series of isotonic coefficients
can be determined. The following table gives a comparison of
the isotonic coefficients of various substances obtained by the
plasmolytic, blood corpuscle and haematocrit methods. The iso-
tonic coefficients are referred to that of cane sugar as unity.

There are other methods which may be used for the comparison
of osmotic pressures, among which may be mentioned that due to
Wladimiroff,t involving the use of bacteria, and the interesting

* Ibid., 17, 164 (1895).

t Zeit. pltfs. Chem., 7, 529 (1891).

206

THEORETICAL CHEMISTRY

Substance.

Plasmolytic
Method.

Corpuscle
Method.

Haematocrit
Method.

C12H22On .

1.00

1 00

MgSO4

1 09

1 27

1 10

KNO8 .

1.67

1 74

1 84

NaCl . .

1 69

1 75

1 74

CHs.COOK

1 67

1 66

1 67

CaCl2

2.40

2,36

2.33

method develbped by Tammann,* in which artificially prepared
membranes are employed.

Osmotic Pressure and Diffusion. That there is a very close
connection between osmotic pressure and diffusion, has already
been pointed out. In fact the osmotic pressure of a solution may
be regarded as the dnvmgTorce which causes the mdle^c^e^of a
dissolved substance to distribute themselves uniformly through-
out the solution.

The process of diffusion was first systematically investigated
by Graham f in 1850, but it was not until five years later that
the general law of diffusion was enunciated by Fick.t He proved
theoretically and experimentally that the quantity of solute, ds,
which diffuses through an area A, in a time dt, when the concen-
tration changes by an amount dc, in a distance dx, at right angles
to the plane of A, is given by the equation

ds= -DA—dt,

dx '

in which D is a constant, known as the coefficient of diffusion.
Interpreting the equation of Fick in words, we see that the coeffi-
cient of diffusion is the amount of solute which will cross 1 square
centimeter in 1 second, if the change of concentration per centi-
meter is unity.
The phenomena of diffusion have also been investigated by

* Wied. Ann., 34. 299 (1888).

t Phil. Trans. (1850), p. 1, 805; (1861), p. 483.

J Pogg. Ann., 94, 59 (1855).

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 207

Nernst * and Planck.f If we have a tall cylindrical vessel con-
taining a solution of a non-electrolyte in its lower part, and pure
water at the top, the solute will slowly diffuse upward into the
water.

Assuming the osmotic pressure at a height x, to be P, and letting
A denote the area of cross-section of the cylinder, the solute in
the layer whose volume is A dx, will be subjected to a force equal
to — A dPy the negative sign indicating that the force acts in the
direction of diminishing pressure. If c is the concentration in
mols per cubic centimeter, the force acting on each molecule in
this layer will be

_ A.^= _!.*?

cA dx c dx

Let F denote the force necessary to drive a single molecule through
the solution with the velocity of one centimeter per second. Since
the velocity is constant, the resistance due to the viscosity of the
medium must also be denoted by F. The velocity attained will
be

_J- <*?

cF' dx

If dN represents the number of molecules crossing each layer in
a time dt, then, since the number crossing unit area per second
is proportional to the concentration and to the mean velocity of
the molecules, we shall have

dN= -~-~
cF dx

or>

,Ar 1 A dP ,.

dN = - nA-j-dt-
F dx

When the solution is dilute, we may apply the general equation,
PV = RT, remembering that V = 1/c. Substituting in the pre-
ceding equation, we have

, RT dc,.

* Zeit. phys. Chem., 2, 40, 615 (1888); 4, 129 (1889).
t Wied. Ann., 40, 561 (1890).

208 THEORETICAL CHEMISTRY

Comparing this equation with that of Fick, we see that the
coefficient of diffusion D, corresponds to the factor, RT/F. From
the equation of Nernst it is possible to calculate the force required
to drive a molecule of solute through the solution with unit veloc-
ity. Thus, solving the above equation for F, we have

By means of this equation, it has been calculated that the force
necessary to drive one molecule of formic acid through water
with a velocity of one centimeter per second at 0° C. is equal to
the weight of 4,340,000,000 kilograms. It is difficult at first to
realize that such enormous forces are operative in solutions, but
when one considers the minute size of the molecules and the great
resistance offered by the medium, it becomes evident that a very
large driving force must be applied to produce an appreciable
movement of the solute through the solution.

Principle of Soret. If a solution is maintained at a uniform
temperature it will ultimately become homogeneous; if, on the
other hand, two parts of a homogeneous solution are kept at
different temperatures for some time, the solution will become
more concentrated in the colder portion. This phenomenon was
first investigated by Soret.* The experiments of Soret are of
special interest, since they furnish a means of determining the
influence of temperature on osmotic pressure. Thus, if the law
of Gay-Lussac holds for osmotic pressure, the colder portion of a
solution should increase in concentration by 1/273 for each degree
of difference in temperature. The experimental results are in
satisfactory agreement with the requirements of theory, and con-
stitute another proof of the applicability of the gas laws to dilute
solutions.

Lowering of Vapor Pressure. It has long been known that
the vapor pressure of a solution is less than that of the pure sol-
vent, provided the solute is non-volatile. The investigations of
von Babo and Wiillner f on the lowering of vapor pressure of

* Ann. Chem. Phys. (5), 22, 293 (1881).
t Pogg. Ann., 103, 529 (1858).

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 209

, ' , ' ' A

various liquids when non-vol^til'e substances are dissolved in
them, resulted in the following generalizations: — (1) The lower-
ing of the vapor pressure of a solution is proportional to the amount
of solute present; and (2) For the same solution, the lowering of the
vapor pressure of the solvent by a non-volatile solute is at all tempera-
tures a constant fraction of the vapor pressure of the pure solvent.

In 1887, Raoult,* as the result of an exhaustive experimental
investigation, enunciated the following laws: — (1) When equi-
mokcular quantities of different non-volatile solutes are dissolved in
equal volumes of the same solvent^ the vapor pressure of the solvent is
lowered by a constant amount; and (2) The ratio of the observed
lowering of the vapor pressure to the vapor pressure of the pure sol-
vent is equal to the ratio of the number of mols of solute to the total
number of mols in the solution. The ratio of the observed lowering
to the original vapor pressure is called the relative lowering of the
vapor pressure. Letting pi and p2 denote the vapor pressures of
solvent and solution, Raoult's second law may be put in the
form

Pi — Pz _ n

in which n and N represent the number of mols of solute and
solvent respectively. Some of Raoult's results for ethereal solu-
tions are given in the accompanying table.

Pl-P2

Mols of Solute

Pi-Pt

Substance.

per 100 mols
of Solution.

for Solution.

for 1 molar
per cent
Solution.

Turpentine

8.95

0.0885

0 0099

Methyl salicylic acid

2.91

0.026

0.0089

Methyl benzoic acid

9.60

0.091

0.0095

Benzoic acid

7.175

0 070

0.0097

Trichioracetic acid

11.41

0.120

0.0105

Aniline

7.66

0.081

0.0106

The results given in the fourth column of the table are nearly

* Compt. rend., 104, 1430 (1887); Zeit. phys. Chem., 2, 372 (1888); Ann.
Chem. Phys. (6), 15, 375 (1888).

210 THEORETICAL CHEMISTRY

constant, and are in close agreement with the theoretical value
of the relative lowering of a 1 molar per cent solution calculated
as follows: —

gL— ~ **
pi

When the solution is very dilute, the number of mols of solute
is negligible in comparison with the number of mols of solvent,
and the equation of Raoult may be written

Pi — Pz n_

Pi N'

Since n = g/m, and N = W/M, where g and W are the weights of
solute and solvent respectively, and m and M are the correspond-
ing molecular weights, the above equation becomes

PI ~ P2 = gM
Pi Wm

This equation enables us to calculate the molecular weight of a
dissolved substance from the relative lowering of the vapor
pressure produced by the solution of a known weight of solute in
a known weight of solvent. Solving the equation for w, we
have

m - gM . Pl .

'** — fir

W pi- pz

As an illustration of the application of this equation, we may
take the determination of the molecular weight of ethyl benzoate
from the following experimental data : — The vapor pressure at
80° C. of a solution of 2.47 grams of ethyl benzoate in 100 grams
of benzene is 742.6 rnm, : the vapor pressure of pure benzene at
80° C. is 751.86 mm. Substituting in the equation, we have

_ 2.47 X 78 751.86

~ 751.86 - 742.6

The molecular weight calculated from the formula, CcHs
is 150.

The difficulties which attend the accurate measurement of the
vapor pressure of a solution by the static method have already

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 211

been mentioned. While there are other methods which are pref-
erable for the determination of the molecular weight of dissolved
substances, the vapor pressure method has one marked advan-
tage, — it can be used for the same solution at widely divergent
temperatures. The method devised by Walker and already
described in connection with the determination of the vapor
pressure of pure liquids (p. 92) is well adapted to the measure-
ment of the vapor pressure of solutions.

Connection between Lowering of Vapor Pressure and Osmotic
Pressure. The relation between osmotic pressure and the
lowering of vapor pressure has been derived
in the following manner by Arrhenius.*
Imagine a very dilute solution contained in
the wide glass tube A> Fig. 64. The tube,

A, is closed at its lower end with a semi-
permeable membrane, and dips into a vessel,

B, which contains the pure solvent. The
entire apparatus is covered by a bell-jar C,
and the enclosed space exhausted. Let
h be the difference in level between the
solvent and solution when equilibrium is
established, that is, when the hydrostatic
pressure of the column of liquid is equal to
the osmotic pressure. When equilibrium is .
attained, the vapor pressure of the solution

at the height "h" will be equal to the pres-
sure of the vapor of the solvent at this height. If the vapor pres-
sure of the pure solvent in the vessel B is pi, and if p2 is the vapor
pressure of the solution at the height h, we shall have

Pi — p* = hd,

where d denotes the density of the vapor. Let v be the volume of
1 mol of solvent in the state of vapor, then

Fig. 64.

and

piv = RT,

v = — •
Pi

* Zeit. phys. Chem., 3, 115 (1889).

212 THEORETICAL CHEMISTRY

If the molecular weight of the solvent is M, we may replace v
by M/d, when the preceding equation becomes

M ^RT

d pi '
or,

The solution being very dilute the osmotic pressure may be cal-
culated from the equation

PV = nRT,

where P is the osmotic pressure of the solution, V the volume
of the solution containing 1 mol of solute, and n the number of
mols of solute present. If s represents the density of the solvent
and also of the solution, since it is very dilute, we may write

P = hs,
and

where g is the number of grams of the solvent in which the n
mols of solute are dissolved. Substituting these values of P and
V in the general equation, we have

PV = nRT = hg,
and solving for A,

h - nRT (to

Q W

Substituting the values of d and h, given in equations (2) and
(3), in equation (Hwe have

nRT Mpi nMpi ,A.

*-»- — "Bf- , ' (4)

Rearranging equation (4), and remembering that N = g/M, we
obtain

Pi — P* __ n ( .

~^ IT (5)

This equation it will be seen, is identical with that derived experi-
mentally by Raoult for very dilute solutions.

DILUTE SOLUTIONS AND OSMOTIC PRESSURES 213

van't Hoff showed, by an application of thermodynamics to
dilute solutions, that the relation between osmotic pressure and
the relative lowering of the vapor pressure is expressed by the
equation

Pi — pz = MP
pi sRT'

in which the symbols have the same significance as above. This
equation may be reconciled easily with the equation of Raoult.
If n in equation (5) be replaced by its equal, PV/RT, the
equation becomes

pi-?>2= PV
pi NET*

But V = NM/s, hence

pi - p2 = MP

pi sRT'

This equation shows that the relative lowering of the vapor pressure
is directly proportional to the osmotic pressure.

Elevation of the Boiling-Point. Just as the vapor pressure of
a solution is less than that of the pure solvent, so the boiling-point
of a solution is correspondingly higher than the boiling-point of
the solvent. It follows that when equimolecular quantities of
different substances are dissolved in equal volumes of the same
solvent, the elevation of the boiling-point is constant. Thus, the
molecular weight of any soluble substance may be determined by
comparing its effect on the boiling-point of a particular solvent,
with that of a solute of known molecular weight. The elevation
in boiling-point produced by dissolving 1 mol of a solute in 100
grams, or 100 cubic centimeters, of a solvent is termed the molec-
ular elevation, or boiling-point constant of the solvent. In deter-
mining the boiling-point constant of a solvent, a fairly dilute
(solution is employed and the elevation in the boiling-point is
observed; the value of the constant is then calculated on the
assumption that the elevation in boiling-point is proportional to
the concentration.

If g grams of a substance of unknown molecular weight m,
are dissolved in W grams of solvent, and the boiling-point is raised

214

THEORETICAL CHEMISTRY

A degrees, then, since m grams of the substance when dissolved
in 100 grams of solvent, produce an elevation of K degrees (the
molecular elevation), it follows that

1000.

therefore,

: A :: m : K,

The accompanying table gives the boiling-point constants for
100 grams and 100 cubic centimeters of some of the more com-
mon solvents.

Solvent.

Molecular Elevation.

100 gr.

100 cc.

Water.
Ethyl alcohol
Ether

5 2
11 5
21 0
16 7
26 7
35 6
30 1

5 4
15 6
30 3
22 2
32 8

Acetone ... .

Benzene . . ...
Chloroform . .
Pyridine . .

As an example of the calculation of the molecular weight of a
dissolved substance by the above formula, we may take the
calculation of the molecular weight of camphor in acetone from
the following data: —

When 0.674 gram of camphor is dissolved in 6.81 grams oi
acetone, the boiling-point of the solvent is raised 1°.09. Substitut-
ing in the formula, we have

m = 100 X 16.7 X °'674

6.81 X 1.09

= 151.

The molecular weight of camphor according to the formula
CioHieO, is 152.

The molecular elevation of the boiling-point can be calculated
by means of the formula,

K = 0.02 T2
w

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 215

T ::

in which T is the absolute boiling-point of the solvent, and w is
the heat of vaporization for 1 gram of the solvent at its boiling-
point. This formula will be derived in a subsequent paragraph
of this chapter. The calculated values
of K are in close agreement with the
values obtained experimentally by Raoult
and others. As an example, the calcu-
lated value of the molecular elevation
for water, the heat of vaporization of
which at 100° C. is 537 calories, is

0.02 X (373)2 = r
537 "" ° '

a value in exact agreement with the exper-
imental value given in the table.

Experimental Determination of Mo-
lecular Weight by the Boiling-Point
Method. One of the simplest and most
convenient of the various forms of appa-
ratus which have been devised for the
determination of the boiling-point of
solutions, is that developed by Jones,*
and shown in Fig. 65. The liquid whose
boiling-point is to be determined is intro-
duced into the vessel A, which already
contains a platinum cylinder P, em-
bedded in some glass beads. Sufficient
liquid is introduced to insure the com-
plete covering of the bulb of the ther-
mometer, as shown in the sketch.

The side tube of A is connected with a condenser, C.

65.

The

vessel A, is surrounded by a thick jacket of asbestos J, and rests
on a piece of asbestos board in which a circular hole is cut, and
over which a piece of wire gauze is placed. The liquid is heated
by means of a burner, B. The platinum cylinder is the feature
which differentiates this apparatus from the various other forms

* Am. Chem. Jour., 19, 581 (1897).

216 THEORETICAL CHEMISTRY

of boiling-point apparatus. It has the two-fold object of pre-
venting the condensed solvent from coming in direct contact with
the bulb of the thermometer, and of reducing the effect of radia-
tion to a minimum. The liquid in A is boiled, using a very small
flame, until the thermometer remains constant; this temperature
is taken as the boiling-point of the liquid. The apparatus is now
emptied and dried. A weighed amount of the liquid is then intro-
duced into A, and to this is added a known weight of solute; the
thermometer is replaced and the boiling-point of the solution is
determined. The difference between the readings of the thermom-
eter when immersed in the solution, and in the solvent alone, gives
the boiling-point elevation. For further details concerning the
boiling-point method as applied to the determination of molecular
weights, the student is referred to any one of the standard labo-
ratory manuals.

Osmotic Pressure and Boiling-Point Elevation. Imagine a
dilute solution containing n mols of solute in G grams of solvent,
and let dT be the elevation in the boiling-point. Suppose a large
quantity of the solution to be introduced into a cylinder, fitted
with a frictionless piston, and closed at the bottom by a semi-
permeable membrane. Let the cylinder and contents be raised
to the absolute temperature T°, the boiling-point of the solvent,
and then let pressure be exerted on the piston just sufficient
to overcome the osmotic pressure of the solution. In this way, let
a quantity of solvent corresponding to 1 mol of solute be forced
through the semi-permeable membrane. The volume V, thus
expelled is the volume corresponding to G/n grams of solvent.
If the osmotic pressure of the solution is P, then the work done in
moving the piston and expelling the solvent is PV. Now let the
portion of the solvent which has been forced through the semi-
permeable membrane be vaporized. For this operation G/n . w
calories will be required, w being the heat of vaporization for 1
gram of solvent at its boiling-point. Then let the entire system
be raised to the boiling-point of the solution (T + dT)°, the pre-
viously expelled G/n grams of vapor being allowed to mix with
the solution. The heat of vaporization lost at T° is thus recov-
ered at the slightly higher temperature, (T + dT)°. Finally, the

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 217

entire system is cooled to T°, and is thus restored to its original
state. Applying the well-known thermodynamic relation, that
the ratio of the work done to the heat absorbed, is the same as
the ratio of the difference in temperature to the absolute initial
temperature of the system, we have

therefore,

But, since PV = RT, equation (1) may be written

If n = I and G = 100 grams, then dT = K (the molecular eleva-
tion of the boiling-point), or

(
(2)

lOOll)

Or putting R — 2 calories, we have

0.02 T*

Equation (1) shows that the osmotic pressure of a solution is directly
proportional to the elevation of the boiling-point. Equation (2) was
originally derived by van't Hoff at about the time when Raoult
determined the values of K experimentally.

Lowering of the Freezing-Point. Of all the methods employed
for the determination of molecular weights in solution, the freez-
ing-point method is the most accurate and the most widely used.
It was pointed out by Blagden * over a century ago, that the de-
pression of the freezing-point of a solvent by a dissolved substance is
directly proportional to the concentration of the solution. When
equimolecular quantities of different substances are dissolved in
equal volumes of the same solvent, the lowering of the freezing-
point is constant. The molecular weight of any soluble sub-

* Phil Trans., 78, 277 (1788).

218

THEORETICAL CHEMISTRY

stance can be found, asrin the beilmg-point method, by comparing
its effect on the freezing-point of a solvent with that of a solute
of known molecular weight. The molecular lowering of the freezing-
point, or the freezing-point constant, of a solvent is defined as the de-
pression of the freezing-point produced by dissolving 1 mol of solute
in 100 grams or 100 cubic centimeters of solvent. The freezing-point
constants of a few common solvents are given in the following table.

Solvent.

Molecular Depression.

100 gr.

100 cc.

Water

18 5
50
74
69
39

18 5
56

41 "

Benzene. ...

Phenol
Naphthalene . . . . . ...
Acetic acid. ..... .

van't Hoff showed that the molecular lowering of the freezing-
point of a solvent K, can be calculated from the absolute freez-
ing-point T7, and the heat of fusion w, for 1 gram of solvent at
the temperature T, by means of the formula

= 0.02 T2
w

This expression is analogous to that which applies to the molec-
ular elevation of the boiling-point. The agreement between the
observed and the calculated values of K is very satisfactory, as
the following calculation for water shows: —

v 0.02 X (273)2

18.6.

It is of interest to note that the calculated value of K for water is
lower than the experimental values originally obtained by Raoult
and others. Subsequent experiments, carried out with greater
care and better apparatus, by Raoult, Abegg and Loomis gave
values in close agreement with that derived theoretically. A
formula analogous to that employed for the calculation of the
molecular weight of a dissolved substance from the elevation it

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 219

produces in the boiling-point of a solvent, may be used for the
calculation of molecular weight from freezing-point depression.
Thus, if g grams of solute when dissolved in W grams of solvent
produce a depression A of the freezing-point of the solvent, the
molecular weight m, is given by the formula

where K is the molecular lowering of the freezing-point.

EXAMPLE. When 1.458 grams of acetone are dissolved in
100 grams of benzene, the freezing-point of the solvent is de-
pressed 1.22°, therefore the molecular weight of acetone is

m = 100 X 50 X j-oo - 59.8.

The molecular weight of acetone, calculated from the formula
C3H60, is 58.

In order to obtain trustworthy results with the freezing-point
method, it is necessary that only the pure solvent separate out
when the solution freezes, and that excessive overcooling be
avoided. When too great overcooling occurs, the subsequent
freezing of the solution results in the separation of so large an
amount of solvent in the solid state, that the observed freezing-
point corresponds to the equilibrium temperature of a more
concentrated solution than that originally prepared. A formula
for the correction of the concentration, due to excessive overcooling,
has been derived by Jones.* If the overcooling of the solution
in degrees be represented by u, the heat of fusion of 1 gram of sol-
vent at the freezing-point by w, and the specific heat of the sol-
vent by c, then the fraction of the solvent which will solidify, /,
may be calculated by the formula,

When water is used as the solvent, c = 1 and w = 80. There-

fore, for every degree of overcooling, the fraction of the solvent

separating as ice will be 1/80, and the concentration of the original

* Zeit. phys. Chem., 12, 624 (1893).

220

THEORETICAL CHEMISTRY

solution is increased by just so much. It is simpler, however, to
apply the correction directly to the freezing-point depression in-
stead of to the concentration.

Experimental Determination of Molecular Weight by the
Freezing-Point Method. The apparatus almost universally em-
ployed for the determination of molecular weights by the freezing-
point method is that devised by Beckmann,*
and shown in Fig. 66. It consists of a thick-
walled test tube A, provided with a side tube,
and fitted into a wider tube Ai, thus surround-
ing A with an air space.

The whole is fitted into the metai cover of
a large battery jar, which is filled with a freez-
ing mixture whose temperature is several de-
grees below the freezing-point of the solvent.

The tube A is closed by a cork stopper,
through which passes the thermometer and
stirrer. The thermometer is generally of the
Beckmann differential type. This instrument
has a scale about 6° in length, each degree
being divided into hundredths; the quantity
of mercury in the bulb can be varied by means
of a small reservoir at the top of the scale, so
that the zero of the instrument can be adjusted
for use with solvents of widely different freez-
ing-points. In carrying out a determination
lg* ' with the Beckmann apparatus, a weighed

quantity of solvent is placed in A, and the temperature of the
refrigerating mixture regulated so as to be not more than 5° below
the freezing-point of the solvent. The tube A is removed from its
jacket, and is immersed in the freezing mixture until the solvent
begins to freeze. It is then replaced in the jacket AI, and the
solvent is vigorously stirred. The thermometer rises during the
stirring until the true freezing-point is reached, after which it
remains constant. This temperature is taken as the "freezing
temperature of the solvent.

* Zeit. phys. Chem., 2, 683 (1888).

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 221

The tube A is now removed from the freezing mixture, and a
weighed amount of the substance whose molecular weight is to
be determined is introduced. When the substance has dissolved,
the tube is replaced in A\ and the solution cooled not more than
a degree below its freezing-point. A small fragment of the solid
solvent is dropped into the solution, which is then stirred
vigorously until the thermometer remains constant. The maxi-
mum temperature is taken as the freezing-point of the solution.
The difference between the freezing-points of solution and
solvent is the depression sought. For further details con-
cerning the determination of molecular weights by the freezing-
point method, the student is referred to a physico-chemical
laboratory manual.

Osmotic Pressure and Freezing-Point Depression. Let dT
be the freezing-point depression produced by n mols of solute in
G grams of solvent, the solution being dilute. Imagine a large
quantity of this solution to be confined within a cylinder fitted
with a frictionless piston, the bottom of the cylinder being closed
by a semi-permeable membrane. Let the cylinder and contents
be cooled to the freezing temperature of the solvent I7, and then
let pressure be applied to the piston until a quantity of solvent
corresponding to 1 mol of solute is forced through the semi-perme-
able membrane. This requires an expenditure of energy equiva-
lent to PV, where P is the osmotic pressure of the solution and V
is the volume of solvent expelled. The volume V is clearly the
volume of 0/n grams of solvent. Now let the expelled portion

S*1

of solvent be frozen and the system deprived of — w calories of

heat, where w is the heat of fusion of 1 gram of the solvent at the
temperature T.

The temperature of the solution is then lowered to its freezing-
point (T — dT)^ and the 0/n grams of solidified solvent dropped
into it. The solidified solvent melts, thereby restoring to the
system at the temperature (T — dT), the heat of fusion formerly
taken from it. Finally, the temperature of the system is raised
to Tf, the initial temperature of the cycle. Applying the familiar
thermodynamic relation, that the ratio of the work done to the

222 THEORETICAL CHEMISTRY

heat absorbed, is the same as the ratio of the difference in temper-
ature to the initial absolute temperature, we have

PV _ dT. ()

G ~T~ W

— w
n

From which we obtain

w G
But PV = RT, hence equation (1) becomes

UJ ~ w G

If n = 1 and G = 100 grams, then dT = K, the molecular lower-
ing of the freezing-point, and

RT*
100 W'

Or putting R = 2 calories, we have

A' = . (2)

w ^ J

An equation to which reference has already been made.

It is evident from equation (1) that the osmotic pressure of a
solution is directly proportional to the freezing-point depression.

Molecular Weight in Solution. As has been pointed out, the
molecular weight of a dissolved substance can be readily calcu-
lated, provided that the osmotic pressure of a dilute solution of
known concentration at known temperature is determined. But
the experimental difficulties attending the direct measurement
of osmotic pressure are so great, that it is customary to employ
other methods based upon properties of dilute solutions which
are proportional to osmotic pressure. We have shown that in
dilute solutions osmotic pressure is directly proportional (1) to
the relative lowering of the vapor pressure, (2) to the elevation
of the boiling-point, and (3) to the depression of the freezing-point.

From this it follows, that equimolecular quantities of different
substances dissolved in equal volumes of the same solvent, exert the

DILUTE SOLUTIONS AND OSMOTIC PRESSURE 223

same osmotic pressure, and produce the same relative lowering of
vapor pressure, the same elevation of boiling-point, and the same
depression of freezing-point. Since equimolecular quantities of
different substances contain the same number of molecules, it is
evident that the magnitude of osmotic pressure, relative lowering of
vapor pressure, elevation of boiling-point and depression of freezing-
point, is dependent upon the number of particles present in the solu-
tion and is independent of their nature. It has been pointed out
by Nernst that any process which involves the separation of
solvent from solute, may be employed for the determination of
molecular weights. A little reflection will convince the reader
that the four methods discussed in this chapter involve such separ-
ation. Both van't Hoff and Raoult emphasized the fact that the
formulas derived for the determination of molecular weights in so-
lution depend upon assumptions which are valid only for dilute
solutions. It follows, therefore, that we are not justified in apply-
ing these formulas to concentrated solutions. Up to the present time
we have no satisfactory theory of concentrated solutions, neither
can we state up to what concentration the gas laws apply.

PROBLEMS.

1. At 10° C. the osmotic pressure of a solution of urea is 500 mm.
of mercury. If the solution is diluted to ten times its original volume,
what is the osmotic pressure at 15° C.? Ans. 50.89 mm.

2. The osmotic pressure of a solution of 0.184 gram of urea in 100 cc.
of water was 56 cm. of mercury at 30° C. Calculate the molecular weight
of urea. Ans. 62.12.

3. At 24° C. the osmotic pressure of a cane sugar solution is 2.51 atmos-
pheres. What is the concentration of the solution in mols per liter?

Ans. 0.103.

4. At 25°. 1 C. the osmotic pressure of solution of glucose containing
18 grams per liter was 2.43 atmospheres. Calculate the numerical value
of the constant R, when the unit of energy is the gram-centimeter.

Ans. 84,231.

Jb. The vapor pressure of ether at 20° C. is 442 mm. and that of a solu-
tion of 6.1 grams of benzoic acid in 50 grams of ether is 410 mm. at the
same temperature. Calculate the molecular weight of benzoic acid in
ether. Ans. 124.

224 THEORETICAL CHEMISTRY

6. At 10° C. the vapor pressure of ether is 291.8 mm. and that of a
solution containing 5.3 grams of benzaldehyde in 50 grams of ether is
271.8 mm. What is the molecular weight of benzaldehyde?

Ans. 106.6.

7. A solution containing 0.5042 gram of a substance dissolved in 42.02
grams of benzene boils at 80°. 175 C. Find the molecular weight of the
solute, having given that the boiling-point of benzene is 80°.00 C.. and its
heat of vaporization is 94 calories per gram. Ans. 181.9.

8. A solution containing 0.7269 gram of camphor (mol. wt. = 152)
in 32.08 grams of acetone (boiling-point = 56°.30 C.) boiled at 56°.55 C.
What is the molecular elevation of the boiling-point of acetone? What is
its heat of vaporization?

Ans. K = 16.74; w = 129.5 cals. per gm.

9. A solution of 9.472 grams of CdI2 in 44.69 grams of water boiled at
100°.303 C. The heat of vaporization of water is 536 calories per gram.
What is the molecular weight of CdI2 in the solution? What conclusion
as to the state of CdI2 in solution may be drawn from the result?

Ans. 363.2.

10. The freezing-point of pure benzene is 5°.440 C. and that of a solu-
tion containing 2.093 grams of benzaldehyde in 100 grams of benzene is
4°.440 C. Calculate the molecular weight of benzaldehyde in the solu-
tion. K for benzene is 50. Ans. 104.6.

11. A solution of 0.502 gram of acetone in 100 grams of glacial acetic
acid gave a depression of the freezing-point of 0°.339 C. Calculate the
molecular depression for glacial acetic acid. Ans. 39.

12. By dissolving 0.0821 gram of m-hydroxybenzaldehyde (C7H602) in
20 grams of naphthalene (melting point 80°. 1 C.) the freezing-point is
lowered by 0°.232 C. Assuming that the molecular weight of the solute
is normal in the solution, calculate the molecular depression for naphtha-
lene and the heat of fusion per gram.

Ans. K = 68.96; w = 36.2 cals. per gm.

CHAPTER XII.
ASSOCIATION, DISSOCIATION AND SOLVATION.

Abnormal Solutes. As has already been pointed out the
acceptance of Avogadro's hypothesis was greatly retarded by the
discovery of certain substances whose vapor densities were ab-
normal. Thus, the vapor density of ammonium chloride is approx-
imately one-half of that required by the formula NH4C1, while
the vapor density of acetic acid corresponds to a formula whose
molecular weight is greater than that calculated from the formula,
C2H4O2. The anomalous behavior of ammonium chloride and
kindred substances <has been shown to be due, not to a failure of
Avogadro's law, but to a breaking down of the molecules — a
process known as dissociation. The abnormally large molecular
weight of acetic acid on the other hand, has been ascribed to a
process of aggregation of the normal molecules, known as asso-
ciation. In extending the gas laws to dilute solutions similar
phenomena have been encountered.

Association in Solution. When the molecular weight of acetic
acid in benzene is determined by the freezing-point method, the
depression of the freezing-point is abnormally small and conse-
quently, as the formula

shows, the molecular weight will be greater than that correspond-
ing to the formula, C2H4O2. Acetic acid in benzene solution is
thus shown to be associated. Almost all compounds containing
the hydroxyl and cyanogen groups when dissolved in benzene are
found to be associated. Solvents, such as benzene and chloro-
form, are frequently termed associating .solvents, although it is
doubtful whether they exert any associating action. There is
considerable experimental evidence to show that those substances

225

226 THEORETICAL CHEMISTRY

whose molecules are associated in benzene and chloroform solu-
tion, are also associated in the free condition. Just as the depres-
sion of the freezing-point of a solution of an associated substance
is abnormally small, so its osmotic pressure and other related
properties will be less than the calculated values.

Dissociation in Solution. Van't Hoff * pointed out that the
osmotic pressure of solutions of most salts, of all strong acids, and
of all strong bases is much greater for all concentrations than
would be expected from the osmotic pressure of solutions of sub-
stances, like cane sugar or urea, for corresponding concentrations.
He was unable to account for this abnormal behavior, and in order
to render the general gas equation applicable, he introduced a
factor it the modified equation being

PV = iRT.

If the osmotic pressure of some substance, like cane sugar, which
behaves normally, be represented by P0, the factor i is given by
the expression

. P

Since the osmotic pressure of a solution is proportional to the
relative lowering of its vapor pressure, to the elevation of its
boiling-point, and to the lowering of its freezing-point, we may
write

. _ ^ Pl

where the symbols have their usual significance. The subscript
0 refers in each case to a substance which behaves normally, and A
denotes either boiling-point elevation or freezing-point depression.
A. more definite conception of the abnormal behavior of salts
will be gained by an inspection of the accompanying tables. In
the first column is recorded the molar concentration of the solu-
tion; the second column gives the observed depressions of the
* Zeit. phys. Chem., i, 501 (1887).

ASSOCIATION, DISSOCIATION AND SOLVATION 227

freezing-point and the third column contains the values of the
ratio of the observed depression to the normal depression, or i.

POTASSIUM CHLORIDE.

POTASSIUM SULPHATE.

•

{

0 05

0 1750

1 88

0 05

0 2270

2 33

0 10

0 3445

1 85

0 10

0 4317

2 32

0 20

0 6808

1 83

0 20

0 8134

2 18

0 40

1 3412

1 80

0 30

1 1673

2 09

ALUMINIUM CHLORIDE.

SODIUM CHROMATE.

0 046
0.076
0 102

0 276
0 446
0 578

3 22
3.15 -
3.04

0.1
0 2
0 5
1 0

0 450
0 850
1.960
3 800

2 42
2 28
2 11
2 04

It is apparent from the above data that the depression of the
freezing-point of water caused by these salts is abnormally large,
a fact which points to an increase in the number of dissolved units
over that corresponding to the initial concentration.

The Theory of Electrolytic Dissociation. In 1887, Arrhenius *
advanced an hypothesis to account for the abnormal osmotic
activity of solutions of acids, bases and salts. He pointed out
that just as the exceptional behavior of certain gases has been
completely reconciled with the law of Avogadro, by assuming a
dissociation of the vaporized molecule into two or more simpler
molecules, so the enhanced osmotic pressure and the abnormally
great freezing-point depression of solutions of acids, bases and
salts can be explained, if we assume a similar process of dissoci-
ation. He proposed, therefore, that aqueous solutions of acids,
bases and salts be considered as dissociated, to a greater or less
extent, into positively- and negatively-charged particles or ions,
and that the increase in the number of dissolved units due to this

* Zeit. phys. Chem., i, 631 (1887).

228 THEORETICAL CHEMISTRY

dissociation is the cause of the enhanced osmotic activity. Accord-
ing to this hypothesis, hydrochloric acid, potassium hydroxide and
potassium chloride, when dissolved in water, dissociate in the
following manner: —

HC1 -»H'+C1'

KC1 -»K' + C1',

where the dots indicate positively-charged ions and the dashes
negatively-charged ions.

In each of the above cases, one molecule yields two ions, so
that, if dissociation is complete, the maximum osmotic effect
should not be greater than twice that produced by an equimolecu-
lar quantity of a substance which behaves normally. Reference
to the preceding table shows that the value of i for potassium
chloride approaches the limiting value of 2 as the solution is
diluted. The other salts given in the table dissociate, according
to Arrhenius, in the following way: —

AlCl3-»Ar'+3Cl'.

If these equations correctly represent the process of dissociation,
then when dissociation is complete, the osmotic effect of infinitely
dilute solutions of potassium sulphate and sodium chromate
should be three times the effect produced by an equimolecular
quantity of a normal solute, and in the case of aluminium chloride,
the maximum effect should be four times the effect due to a normal
substance. A glance at the table shows that the values of i for
solutions of the three salts approach these respective limits. If
this hypothesis of ionic dissociation be accepted, then it becomes
possible to calculate the degree of ionization in any solution by
comparing its freezing-point depression with the freezing-point
lowering of an equimolecular solution of a normal substance.

Let us suppose that the degree of dissociation of 1 molecule of
a dissolved substance is a, each molecule yielding n ions. Then

ASSOCIATION, DISSOCIATION AND SOLVATION 229

1 — a will be the undissociated portion of the molecule, and the
total number of dissolved units will be

1 — a + na.

If A is the depression of the freezing-point produced by the sub-
stance, and AO the depression produced by an equimolecular
quantity of an undissociated substance, then

1 — a + na __ A __ .

i "~ "7T" — l>

i- 1

n- I

It will be observed that this formula is identical with that derived
for the degree of dissociation of a gas (p. 91). If this formula
be applied to the freezing-point data for solutions of potassium
chloride given in the preceding table we find the following per-
centages of dissociation corresponding to the different concentra-
tions:—

POTASSIUM CHLORIDE.

0.05

0 1750

1.88

0.10

0 3445

1.85

0 20

0 6808

1 83

0 40

1 3412

1.80

Thfc figures in the last column show that the degree of dissocia-
tion increases as the concentration diminishes. It was further
pointed out by Arrhenius that all of the substances which exhibit
abnormal osmotic effects, when dissolved in water, yield solutions
which conduct the electric current, whereas, aqueous solutions of
such substances as cane sugar, urea and alcohol, exert normal
osmotic pressures, but do not conduct electricity any better than
the pure solvent. In other words, only electrolytes * are capable of
undergoing ionic dissociation; hence Arrhenius termed his hypoth-
esis the electrolytic dissociation theory. As we have seen, when
potassium chloride is dissolved in water, it is supposed to dissoci-

* The term electrolyte strictly refers to the solution of an ionized substance,
although it is often applied to acids, bases and salts because, when dissolved,
they produce electrolytes. To avoid confusion, the term "ionogen" (ion
former) has been proposed for those substances which give conducting solu-
tions.

230

THEORETICAL CHEMISTRY

ate into positively-charged potassium ions and negatively-charged
chlorine ions. Accordingly when two platinum electrodes, one
charged positively and the other negatively, are introduced into
the solution, the potassium ions move toward the negative elec-
trode and the chlorine ions move toward the positive electrode, the
passage of a current through the solution consisting in the ionic
transfer of electric charges. Since the undissociated molecules,
being electrically neutral, do not participate in the transfer of
electric charges, it follows that the conductance of a solution of
an electrolyte is dependent upon the degree of dissociation. The
relation between electrical conductance and the degree of ioniza-
tion will be discussed in a subsequent chapter. It may be stated
at this point, however, that the values of a based upon measure-
ments of electrical conductance, while showing some discrepancies
in individual cases, are in general in good agreement with the
values obtained by the freezing-point method. Furthermore, the
values of a obtained from freezing-point measurements are in har-
mony with those based upon De Vries' isotonic coefficients.
It will be seen, on referring to the table of isotonic coefficients
(p. 181), that solutions of electrolytes show enhanced osmotic
activity. Thus, the osmotic pressures of equi-molecular solutions
of cane sugar, potassium nitrate and calcium chloride are to each
other as 1 : 167 : 2.40.

The following table illustrates the general agreement between
the values of i calculated, (a) from electrical conductance, (b)
from freezing-point depression, and (c) from De Vries' isotonic
coefficients.

Substance.

Molar Cone.

(a)

(b)

fo)

KC1

0 14

1 86

1 82

1 81

LiCl

0 13

1 84

1.94

1.92

Ca(NO3)2

0.18

2.46

2 47

2.48

MgCU

0 19

2.48

2.68

2.79

CaCl2

0.184

2.42

2.67

2.78

It must be remembered that the values of i derived from freez-
ing-point measurements correspond to temperatures in the

ASSOCIATION, DISSOCIATION AND SOLVATION 231

neighborhood of 0° C., while those derived from the other
methods correspond to temperatures ranging from 17° C. to
25° C.

Chemical Properties of Completely Ionized Solutions. The

chemical properties of an ion are very different from the properties
of the atom or radical when deprived of its electrical charge.
For example, the sodium ion is present in an aqueous solution of
sodium chloride, but there is no evidence of chemical reaction with
the solvent; whereas, the element in the electrically-neutral con-
dition reacts violently with water, evolving hydrogen and form-
ing a solution of potassium hydroxide. Again, take the element
chlorine: when chlorine in the molecular condition, either as gas
or in solution, is added to a solution of silver nitrate, no precipitate
of silver chloride is formed. Further, chlorine in such compounds
as dlVls, OClt, etc., is not precipitated by silver nitrate, since
these compounds are not dissociated by water. Or, chlorine may
bo present in a compound which is dissociated by water and yet
not exhibit its characteristic reactions, because it is present in a
complex ion. Thus, potassium chlorate dissociates in the follow-
ing manner: — -

On adding silver nitrate, there is no precipitation, because the
chlorine forms a complex ion with oxygen.

Physical Properties of Completely Ionized Solutions. The
physical properties of completely ionized solutions are, in general,
additive. This is well illustrated by a series of solutions of
colored salts, the color of which is due to the presence of a par-
ticular ion. It is found, when the solutions are sirfficiently
dilute to insure complete dissociation, that they all have the
same color. The additive character of the colors of solutions of
electrolytes is brought out in a striking manner by a comparison
of their absorption spectra. Ostwald * photographed the absorp-
tion spectra of solutions of the permanganates of lithium, cadmium,
ammonium, zinc, potassium, nickel, magnesium, copper, hydrogen
and aluminium, each solution containing 0.002 grain-equivalents

* Zcit. phys. Chein., 9, 579 (1892).

232 THEORETICAL CHEMISTRY

of salt per liter. The absorption spectra, as shown in Pig. 67,
will be seen to be practically identical, the bands occupying the
same position in each spectrum. This affords a strong con-
firmation of the theory of electrolytic dissociation, according to

which a dilute solution is to be
regarded as a mixture of elec-
trically equivalent quantities of
oppositely charged ions, each of
which contributes its specific
properties to the solution. The
permanganate ion being colored,
and common to all of the salts
examined, and the positive ions
of the various substances being
colorless, it follows that when dis-
sociation is complete, the absorp-
tion spectra of all of the solutions
must be identical. A number of
other properties of completely
dissociated solutions have been
shown to be additive. Among
these may be mentioned density,
specific refraction, surface ten-
sion, thermal expansion, and
magnetic rotatory power. Addi-
tional evidence in favor of the
theory of electrolytic dissociation
will be furnished in forthcoming
.p. 67 chapters. Notwithstanding the

large number of facts which can
be satisfactorily interpreted by the theory, there are directions
in which it requires amplification and modification. Of the
various objections which have been urged against the theory of
electrolytic dissociation, one is of sufficient weight to call for
brief consideration here. When two elements, such as potassium
and chlorine, Combine to form potassium chloride, the reaction
is violent and a large amount of heat is developed. Nevertheless,

ASSOCIATION, DISSOCIATION AND SOLVATION 233

according to the ionization theory, the strong, mutual affinity of
these two elements is overcome by the act of solution in water,
the molecule being split into two oppositely-charged atoms. Obvi-
ously such a separation calls for the expenditure of a large amount
of energy, and the question naturally arises: — What is the
source of this energy? While this question cannot be fully
answered here, it may be pointed out that we have abundant
evidence to show that the ions are hydrated, each being surrounded
by an "atmosphere" of solvent. In view of this fact, it has been
suggested * that dissociation in aqueous solution is caused by the
mutual attraction between the ions and the molecules of the sol-
vent, the heat of ionic hydration furnishing the energy necessary
for the separation of the ions.

Freezing-Point Depressions Produced by Concentrated Solu-
tions of Electrolytes. As has already been mentioned, the
dissociation of electrolytes in aqueous solution increases with the
dilution, becoming complete at a concentration of about 0.001
molar. We should expect the dissociation to diminish with in-
creasing concentration, until, if the electrolyte is sufficiently solu-
ble, the depression of the freezing-point becomes normal. Recent
investigations by Jones and his co-workers f have shown that the
facts are contradictory to this expectation. They found that the
value of the molecular depression of the freezing-point of water
produced by a number of chlorides and bromides, diminished with
increasing concentration up to a certain point, as would be ex-
pected, and then increased again. The increase in the molecular
depression became very marked at great concentrations; in fact,
the molecular depression in a molar solution was frequently greater
than the molecular depression corresponding to a completely
dissociated salt. This phenomenon was systematically studied
by Jones and the author J and the fact was established that it
is quite general.

* Trans. Faraday Soc., i, 197 (1905); 3i 123 (1907).

t Am. Chem. Jour., 22, 5, 110 (1899); 23, 89 (1900).

i Zeit. phys. Chem., 46, 244 (1903); Phys. Rev., 18, 146 (1904); Am. Chem.
Jour., 31, 303 (1904); 32, 308 (1904); 33, 534 (1905); 34, 291 (1905); Zeit,
phys. Chem., 49, 385 (1904); Monograph No. 60, Carnegie Institution of
Washington.

234 THEORETICAL CHEMISTRY

This abnormal depression of the freezing-point may be accounted
for by assuming that the dissolved substance has entered into com-
bination with a portion of the water, thus removing it from the
role of solvent. The formation of a loose molecular complex be-
tween one molecule of the solute and a large number of molecules
of water, acts as a single dissolved unit in depressing the freezing-
point of the pure solvent. Evidently the total amount of water
present, which functions as solvent, is diminished by the amount
of water which has been appropriated by the solute. The abnor-
malities observed in the depression of the freezing-point of con-
centrated solutions of electrolytes can be explained by assuming
that the molecules of solute, or the resulting ions, are in combina-
tion with a number of molecules of solvent. This hypothesis is
termed the solvate theory, and the loose molecular complexes are
called solvates. Since the solvate theory was first proposed, con-
siderable evidence has been accumulated to confirm its correct-
ness. Reference has already been made to the work of Philip on
the solubility of gases in saline solutions, from which he concludes
that the dissolved salts enter into combination with a portion of
the solvent. The experiments of Morse and the Earl of Berkeley
on osmotic pressure, also seem to point to the solvation of the
dissolved substance.

PROBLEMS.

1. At 18° C, a 0.5 molar solution of NaCl is 74.3 per cent dissociated.
What would be the osmotic pressure of the solution in atmospheres at
18° C.? Am. 20.79.

2. A solution containing 3 mols of cane sugar per liter was found by
the plasmolytic method to be isotonic with a solution of potassium nitrate
containing 1.8 mols per liter. What is the degree of ionization of the
potassium nitrate? Ans. 67 per cent.

3. The vapor pressure of water at 20° C. is 17.406 mm. and that of a
0.2 molar solution of potassium chloride is 17.296 mm. at the same tem-
perature. Calculate the degree of dissociation of the salt.

Ans. 75.38 per cent.

4. The degree of dissociation of a 0.5 molar solution of sodium chloride
at 25° is 74.3 per cent. Calculate the osmotic pressure of the solution at
the same temperature. Ans. 21.32 atmos.

ASSOCIATION, DISSOCIATION AND SOLVATION 235

5. A solution containing 1.9 mols of calcium chloride per liter is isotonic
with a solution of glucose containing 4.05 mols per liter. What is the
degree of ionization of the calcium chloride? Ans. 56.6 per cent.

6. At 0° C. the vapor pressure of water is 4.620 mm. and of a solution
of 8.49 grams of NaN03 in 100 grams of water 4.483 mm. Calculate the
degree of ionization of NaNOs. Ans. 64.9 per cent.

7. At 0° C. the vapor pressure of water is 4.620 mm. and that of a
solution of 2.21 grams of CaCl2 in 100 grams of water is 4.583 mm. Cal-
culate the apparent molecular weight and the degree of ionization of
CaCl2. Ans. M = 49.66, a = 62 per cent.

8. The boiling-point of a solution of 0.4388 gram of sodium chloride
in 100 grams of water is 100°.074 C. Calculate the apparent molecular
weight of the sodium chloride and its degree of ionization. K = 5.2.

Ans. M = 30.84, « = 89.7 per cent.

9. The boiling-point of a solution of 3.40 grams of BaCl2 in 100 grams
of water is 100°.208 C. K = 5.2. What is the degree of ionization of
the BaCl2? Ans. 72.5 per cent.

10. At 100° C. the vapor pressure of a solution of 6,48 grams of ammo-
nium chloride in 100 grams of water is 731.4 mm. K = 5.2. What is the
boiling-point of the solution? Ans. 101°,086 C. ^

11. A solution of 1 gram of silver nitrate in 50 grams of water freezes
at -0°.348C. Calculate to what extent the salt is ionized in solution.
K = 18.6. Ans. 59 per cent.

12. A solution of NaCl containing 3.668 grams per 1000 grams of
water freezes at -0°.2207 C. Calculate the degree of ionization of the
salt. K = 18.6. Ans. 89.2 per cent.

13. The freezing-point of a solution of barium hydroxide containing
1 mol in 64 liters is -0°.0833 C. What is the concentration of hydroxyl
ions in the solution? Take K = 18.9 for concentrations in mols per
liter. Ans. 0.0284 gm.-ion per liter.

14. The vapor*pressure of water at 0° C. is 4.620 mm., and the lower-
ing of the vapor pressure produced by dissolving 5.64 grams of sodium
chloride in 100 grams of water is 0.142 mm. What is the freezing-point
of the solution? K » 18.6. Ans. -3°.177 C.

15. A solution containing 8.34 grams Na2S04 per 1000 grams of water
freezes at — 0°.280 C. Assuming dissociation into 3 ions, calculate the

236 THEORETICAL CHEMISTRY

degree of ionization and the concentrations of the Na* and S(V' ions.
K - 18.6.

Am. a — 78.2 per cent; cone. Na* = 0.0918 grrn.-ion per liter; cone.
SO4" - 0.0459 gm.-ion per liter.

CHAPTER XII.
COLLOIDS.

Crystalloids and Colloids. In the course of his investigations
on diffusion in solutions, Thomas Graham * drew a distinction
between two classes of solutes, which he termed crystalloids and
colloids. Crystalloids, as the name implies, can be obtained in the
crystalline form : to tliis class belong nearly all of the acids, bases
and salts. Colloids, on the oilier hand, are generally amorphous,
such substances as albumin, starch and caramel being typical of
the class. Because of the gelatinous character of many of the sub-
stances in this class, Graham termed them colloids (*o\Xa = glue,
and erdos = form). The differences between the two classes are
most apparent in the physical properties of their solutions. Thus,
crystalloids diffuse much more rapidly than colloids; the velocity
of diffusion of caramel being nearly 100 times slower than that of
hydrochloric acid at the same temperature. While crystalloids
exert osmotic, pressure, lower the vapor pressure and depress the
freezing-point of the solvent, colloids have very little effect upon
the properties of the solvent. The marked differences in the rates
of diffusion of crystalloids and colloids render their separation
comparatively easy. If a solution containing both crystalloids
and colloids be placed in a vessel over the bottom of which is
stretched a colloidal membrane, such as parchment, and the whole
is immersed in pure water, the crystalloids will pass through the
membrane, while the colloids will he left behind. This process
was termed by Graham, dialysis, while the apparatus employed to
effect such a separation was called a dialyzer. When a solution of
sodium silicate is added to an excess of hydrochloric acid, the
products of the reaction, silicic acid and sodium chloride, remain
in solution. When the mixture is placed in a dialyzer, the sodium,
chloride and the hydrochloric acid, being crystalloids, diffuse

* Lieb. Ann., 121, 1 (1862).
237

238 THEORETICAL CHEMISTRY

through the membrane of the dialyzer, leaving behind the colloidal
silicic acid.

The terms crystalloid and colloid, as used at the present time,
have acquired different meanings from those assigned to them by
Graham. The terms arc now considered to refer, not so much to
different classes of substances, as to different states which almost
all substances can assume under certain conditions.

Colloidal Solutions. A colloidal solution is OIK* in which the
solute is a colloid, although the latter may not he included among
the substances classified as such by Graham. For example,
arsenious sulphide, ferric hydroxide or finely-divided gold may
form colloidal solutions. In bringing such substances into the
colloidal state, mere agitation with water will riot suffice, but some
indirect method must be employed.

Nomenclature. Graham distinguished between two condi-
tions in which colloids were obtainable, the term sol being applied
to forms in which the system resembled a liquid, while the term
gel was used to designate those forms which were solid arid jelly-
like. When one of the components of the solution was water, the
two forms were called a Jiydrosol and a Jn/drogcL In like manner,
when alcohol was one of the components, the terms alcosol and
alcogel were applied to the two forms.

As the knowledge of colloids has developed it has become neces-
sary to supplement Graham's nomenclature by the introduction
of various other terms. It is known to-day that the essential
difference between colloidal suspensions and solutions on the one
hand, and true solutions on the other, is due to the difference in the
degree of subdivision or degree of dixpersity of the dissolved sub-
stance. In a true solution the dissolved substance is generally
present either in the molecular or ionic condition, as may be shown
by means of the familiar osmotic methods for molecular weight de-
termination. In colloidal solutions, however, the degree of clis-
persity is not so great and has been found to vary from above the
limit of microscopic visibility (1 X 10~5 cm.) to that of molecular
dimensions (1 X 10~~8 cm.). When the degree of dispersity varies
from 1 X 10~3 cm. to 1 X 10~5 cm. the particles are termed
microns. The properties of the disperse phase at this degree of

COLLOIDS 239

dispersity differ appreciably from the properties of the same sub-
stance when present in large masses. When the degree of dis-
persity lies between 1 X 10~5 cm. and 5 X 10~7 cm. the particles
are known as submicrons. The existence of particles whose diam-
eters are approximately 1 X 10~7 cm. has been demonstrated by
Zsigmondy with the ultramicroscope; these minute particles are
termed amicrons. When the degree of dispersity is increased
beyond this limit all heterogeneity apparently vanishes and we
enter the realm of true solutions.

When the dispersion is not too great, colloidal solutions may be
divided into suspensions and emulsions according to whether the
disperse phase is a solid or a liquid. As the dispersion is increased
we obtain suspension and emulsion colloids which may be con-
veniently called suspensoids and emulsoids. Suspensoids and
emulsoids are included under the general term dispersoids. In
certain cases, although the disperse phase is unquestionably liquid,
the systems resemble suspensoids in their behavior, while in other
cases, where the disperse phase is solid, the systems exhibit proper-
ties characteristic of emulsoids. For this reason the classification
of sols as suspensoids and emulsoids is not entirely satisfactory.
A better system is that in which the presence or the absence of
affinity between the disperse phase and the dispersion medium is
made the basis of classification. Where there is marked affinity
between the two phases, the system is termed lyophile, and where
such affinity is absent, the system is termed lyophobe. When the
dispersion medium is water, the terms hydrophile and hydrophobe
are employed.

In the reversible transformation of a sol into a gel, we are
not warranted in referring to the change from gel to sol as an
act of solution, for if the gel really dissolved, a solution and
not a sol would result. Various terms have been proposed for
these reversible transformations but perhaps the most satisfac-
tory are the terms gelation and solation, the former designating
the formation of a gel from a sol and the latter the reverse
process.

Lyotrope Series. The differences between lyophile and lyo-
phobe sols are frequently very marked, this being especially true

240 THEORETICAL CHEMISTRY

of their behavior toward chemical reagents. The action of chemi-
cal reagents on lyophobe sols is almost wholly confined to the
disperse phase, while the addition of reagents to lyophile sols fre-
quently produces a more marked effect on the dispersion medium
than on the disperse phase. It should be observed that the physi-
cal properties of a lyophobe sol and of the pure dispersion medium
are practically identical, while exactly the reverse is true of lyo-
phile sols. It is well known that the addition of a foreign sub-
stance to a reaction-mixture frequently exerts a marked influence
on the speed of the reaction, notwithstanding the fact that the
nature of the added substance may be such as to render its partici-
pation in the reaction highly improbable. If a series of reagents
are arranged in the order of their influence on a particular reaction,
it has been found that the same sequence is preserved when the
same reagents are added to other reactions of widely different
character. In some reactions the reagents may produce effects
directly opposite to those which they produce in others, but the
sequence remains unchanged. For example, when the same re-
action takes place either in an acid or in an alkaline medium, the
substances which promote the reaction when the medium is acid,
retard it when the medium is alkaline, but the sequence of the
added substances remains the same under both conditions. These
facts make it appear highly probable that the effects produced by
the addition of foreign substances to a chemical reaction are to be
ascribed to the changes which they produce in the pure solvent.
It is not without significance that the same sequence of reagents is
maintained whether we observe their influence on different chemi-
cal reactions or on certain physical properties of the solvent, such
as its density, viscosity, and surface tension.

The following examples * afford a striking illustration of the
persistence of the sequence of reagents, generally known as the
lyotrope series. The ions which precede the formula (H^O) reduce
the velocity of reaction or cause a diminution in the magnitude of
the particular physical property tabulated. The ions which follow
the formula (H20) exert the opposite effect.

* Freundlich, Kapillarchemie, p. 411.

COLLOIDS 241

1. The hydrolysis of esters by acids.

Anions: S04 (H2O) Cl < Br.
'Cations: (H2O) Li < Na < K < Rb < Cs.

2. The hydrolysis of esters by bases.

Anions: I > N03 > Br > Cl (H20) S203 < S04.
Cations: Cs > Rb > K > Li (H2O).

3. The viscosity of aqueous solutions.

Anions: NO3 > Cl (H20) S04 (potassium salts).
Cations: Cs > Rb > K (H20) Na < Li (chlorides).

4. The surface tension of aqueous solutions.

(H20) I< N03 < Cl< S04 < C03.

It will be observed that although the ions which accelerate the
acid hydrolysis retard the basic hydrolysis, the sequence of the
ions nevertheless remains the same.

The Ultramicroscope. When a narrow beam of sunlight is
admitted into a darkened room, the dust particles in its path are
rendered visible by the scattering of the light at the surface of the
particles. If the air of the room is free from dust, no shining
particles will be seen and the space is said to be " optically void."
When the particles of dust are very minute, the beam of light
acquires a bluish tint. The blue color of the sky is thus attrib-
uted to the presence of extremely fine particles of dust in the air
together with minute drops of condensed gases in the upper regions
of the atmosphere.

The visibility of a beam of light due to the scattering effect of
minute particles, is known as the Tyndall phenomenon. Almost
all colloidal solutions exhibit this phenomenon when a powerful
beam of light is passed through them, thus proving the presence
of discrete particles in the solutions.

The ultramicroscope is an instrument devised by Siedentopf
and Zsigmondy * for the detection of colloidal particles much too
small to be seen by the naked eye. A powerful beam of light
issuing from a horizontal slit is brought to a focus within the
colloidal solution under examination by means of a microscope
objective, and this image is viewed through a second micro-

* "Colloids and the Ultramicroscope," by R. Zsigmondy. Trans, by
Alexander, John Wiley & Sons, Inc.

242 THEORETICAL CHEMISTRY

scope, the axis of which is at right angles to the path of the
beam.

When examined in this way a colloidal solution appears to be
swarming with brilliantly colored particles, moving rapidly in a
dark field; whereas a true solution if properly prepared, appears
optically void. With the ultramicroscope it is possible to count
the number of particles present in a given volume of a col-
loidal solution. By means of a chemical analysis, the mass of
colloid per unit of volume can be determined and from this the
average mass of each particle can be calculated. If the particles
be assumed to be spherical in shape and to have the same density
as larger masses of the same substance, we can calculate the
volume of a single particle and from this its diameter. Thus,
Burton * in his experiments on gold, silver and platinum sols, found
the average diameter of the colloidal particles to range from 0.2 to
0.6 micron.

Zsigmondy's latest ultramicroscope, Fig. 68, consisting of two
compound microscopes placed at right angles and having their
objectives so cut away as to permit them to be brought together
in focus, enables the observer to discern particles whose diameters
range from 1 to 2 rnilli-microns.

The ultramicroscopic character of emulsoids is by no means
sharply defined, notwithstanding the fact that they exhibit the
Tyndail phenomenon. It has been suggested by Zsigmondy that
the lack of sharpness in definition observed with emulsoids is
probably due to the relatively small difference between the re-
fractive indices of the disperse phase and the dispersion medium.
Where the difference between the refractive indices of the two
phases is very great, as in the case of the metallic sols, excellent
definition is obtained. It is of interest to note that although the
basic hydroxides are apparently suspensoids, yet in their ultra-
microscopic characteristics they closely resemble emulsoids.

Ultrafiltration. Almost all sols can be filtered through ordinary
filter paper without undergoing more than a slight change in con-
centration due to initial adsorption. The rate of filtration varies
widely, depending upon the viscosity of the sol. As a general rule,

* Phil. Mag., ii, 425 (1906).

COLLOIDS

243

emulsoids filter more slowly than suspensoids owing to the high
viscosity of the former.

Fig. 68.

By filtering an arsenious sulphide sol through a porous earthen--
ware filter, Linder and Picton * succeeded in obtaining four differ-

* Jour. Chem. Soc., 61, 148 (1892).

244 THEORETICAL CHEMISTRY

ent sizes of particles which they described as follows: — (1) visible
under the microscope, (2) exhibited the Tyndall phenomenon, (3)
retained by porous plate, and (4) passed through porous plate un-
changed. By employing plates of different degrees of porosity and
determining the average size of the pores which just permit filtra-
tion, it is possible to determine the size of the particles which con-
stitute the disperse phase of a sol. If we make use of a series of
graduated filters, prepared by impregnating filter paper with a
solution of collodion in acetic acid, it is not only possible to sepa-
rate suspensoids from their dispersion media, but also to effect the
concentration of emulsoids. Furthermore such filters are useful
in removing impurities from sols, the impurity passing through the
filter in a manner similar to the passage of the solvent through the
membrane in the process of dialysis. Ultrafiltration is an exceed-
ingly complex process involving the phenomena of adsorption and
dialysis in addition to the ordinary process of mechanical separation.
The complexity of the process is well illustrated by the phenomena
attendant upon the filtration of almost any positive hydrosol.
Thus, if we attempt to filter a ferric hydroxide hydrosol through a
porous plate, or even through an ordinary filter paper, we shall
find that the colloid will be partially retained by the filter. This is
due to the fact that the filter becomes negatively charged in con-
tact with water and, on the entrance of the positively-charged sol
into the pores of the filter, the colloid is immediately discharged
and the disperse phase precipitated. After the pores of the filter
become partially stopped with particles of the colloid, the sol will
then pass through unchanged.

^ Classification of Dispersoids. There is abundant evidence in
favor of the view that colloidal solutions and simple suspensions
are closely related. Suspensions of all grades exist, from those in
which the suspended particles are coarse-grained and visible to
the naked eye, down to those in which a high-power microscope
is required to render the suspended particles visible. Colloidal
solutions have also been shown to be non-homogeneous, the
presence of discrete particles being revealed by means of the
ultramicroscope. It follows, therefore, that the size of the particles
in solution determines whether a substance is to be considered as

COLLOIDS

245

a colloid or not. At one extreme we have true solutions in which
no lack of homogeneity can be detected, even by the ultramicro-
scope, and at the other extreme we have coarse-grained suspen-
sions, in which the particles are visible to the naked eye. Between
these two limits all possible degrees of subdivision are possible
and it is a very difficult matter to draw sharp 1 nes of distinction
between true solutions and colloidal solutions on the one hand,
and between colloidal solutions and suspensions on the other.
One of the most satisfactory schemes of classification is that of
von Weimarn and Wo. Ostwald.* Because of the fact that sus-
pensions, colloidal solutions, and true solutions represent varying
degrees of dispersion of the solute, all three types of system are
termed by these authors, dispersoids. The dispersoids are classi-
fied as shown in the accompanying diagram.

DISPERSOIDS.

Coarse disp

versions (sus-

pensions,

emulsions,

etc.).

Magnitude

of particles

greater than 0.1 /*.*

Colloidal solutions. Molecular

Ionic

Magnitude of particles dispersoids. dispersoids.
between 0.1 /* and 1 /*/*. ' • '

Magnitude of particles,
about 1 nfjL and smaller.

Decreasing degree of

"colloidity."

• ' • • • -• — »

Increasing degree of dis-
persion.

* 1 n = 1 micron - 0.001 mm.

Density of Colloidal Solutions. As we have seen, suspensoids
are commonly regarded as sols in which the disperse phase is solid,
while emulsoids are considered to be sols in which the disperse
phase is liquid. While this distinction between the two classes of
sols is generally well defined, it should be borne in mind that there

* Koll. Zeitschrift, 3, 20 (1908).

246

THEORETICAL CHEMISTRY

are colloidal solutions in which it is extremely difficult to determine
the physical state of the disperse phase.

The fundamental difference between suspensoids and emulsoids
manifests itself most clearly in those properties which undergo
appreciable change in consequence of solution. Among these may
be mentioned density, viscosity and surface tension.

It was shown by Linder and Picton * that the density of
suspensoids follows the law of mixtures. This is clearly shown
by the following table in which are given the observed and
calculated values of the density of a series of arsenious sulphide
sols.

DENSITY OF ARSENIOUS SULPHIDE SOLS.

As2S3 (grams per
liter).

Density (obs.).

Density (calc.).

1.033810

1 033810

1 016880

1 016905

1 008435

1 008440

2 45

1 002110

1 002100

0 1719

1.000137

1.000134

The density of emulsoids, on the other hand, cannot be calcu-
lated from the composition of the sol. This fact may be taken as
evidence in favor of the view that a closer relation exists between
the disperse phase and the dispersion medium in emulsoids than in
suspensoids.

Viscosity of Colloidal Solutions. Owing to the fact that the
concentrations of most suspensoids are relatively small, it follows
that their viscosities differ but little from the viscosity of the pure
dispersion medium. In general, it may be said that the viscosity
of suspensoid sols is slightly greater than that of the dispersion
medium.

On the other hand, the viscosity of emulsoid sols is frequently
much greater than that of the pure dispersion medium. The vis*
cosity of emulsoids also increases with increasing concentration,
as is shown by the data of the following tabLa.

* Jour. Chem. Soc., 67, 71

COLLOIDS

247

VISCOSITY OF EMULSOIDS.

' So..

Temp. 20°
Concentration .

Viscosity.

Per cent

Gelatine ....

0 021

Gelatine ....

0 037

Silicic acid. . . .

0 81

0 012

Silicic acid ....

0 99

0 016

Silicic acid. . . .

1.96

0 032

Silicic acid. . . .

3 67

0 165

Viscosity of water at 20° = 0.0120.

Surface Tension of Colloidal Solutions. The surface tension
of suspensoid sols has been shown by Linder and Picton to be
practically identical with that of the dispersion medium.

As a general rule, the surface tension of emulsoid sols is appre-
ciably smaller than that of the dispersion medium. According to
Quincke * the surface tensions of gelatine sols are appreciably less
than the surface tension of the dispersion medium. The differ-
ence between suspensoids and emulsoids in respect to surface
tension undoubtedly accounts for the fact that, in general, the
former are not adsorbed while the latter are.

Osmotic Pressure of Colloidal Solutions. The osmotic pres-
sure of colloidal solutions is very small. This is what we should
expect with solutions of substances which exhibit a slow rate of
diffusion. As has been pointed out, diffusion is closely connected
with osmotic pressure; hence, if the rate of diffusion is slow, the
osmotic pressure exerted by the solution should be small. In
some cases the osmotic pressure is so small as to escape detection.
The experimental determination of the osmotic pressure of col-
loidal solutions is complicated by the difficulty of removing the
last traces of electrolytes from the colloid. Owing to their great
osmotic activity, the presence of the merest trace of electrolytes
may mask the true osmotic effect of the colloid. It should be borne
in mind, however, that semipermeable membranes are much less
permeable by colloids than by electrolytes and, in consequence of
this fact, the impurities in the colloid would be gradually removed
by prolonged dialysis. If the total osmotic pressure were due to
* WiecL Ann,, III, 35, 582 (1885).

248

THEORETICAL CHEMISTRY

the presence of small amounts of impurities in the colloid, then as
these are removed, the pressure should steadily diminish and ulti-
mately become zero. As a matter of fact, the final value of the
osmotic pressure of a colloidal solution, although generally very
small is never zero. This final, positive value of the osmotic
pressure has been shown to be wholly independent of the method
of preparation of the sol. Although differences in the method of
preparation may introduce different impurities which give rise to
different initial values of the osmotic pressure, in each case the
same final value is obtained. Of course it must be admitted that
the possibility exists that a minute portion of electrolyte which
cannot be removed by dialysis is retained by the colloid, but even
then it is difficult to account for the constancy of the final value of
the osmotic pressure irrespective of the method of preparation of
the sol. The values of the osmotic pressure of suspensoids are
invariably small and by no means concordant.

The following table gives the results obtained by Duclaux * with
colloidal solutions of ferric hydroxide.

OSMOTIC PRESSURE OF COLLOIDAL FERRIC HYDROXIDE.

Cone. Fe(OH)3
Per cent.

Pressure in cm.
of Water.

1 08

0 8

2 04

2 8

3 05

5 6

5 35

12 5

8 86

22 6

Inspection of the table shows that, even in the most concen-
trated solution, the osmotic pressure is very small. Furthermore,
it is apparent that although the osmotic pressure increases with
the concentration, the variables are not proportional. Observa-
tions on the variation of the osmotic pressure of colloidal solu-
tions with temperature, show that, in general, as the temperature
is raised the pressure increases at a more rapid rate than that
required by the law of Gay-Lussac.

Employing membranes of collodion and parchment paper,

* Compt. rend., 140, 1544 (1905); Jour. Chim. Phys., 7, 405 (1909).

COLLOIDS 249

Lillie * and others have demonstrated that the values of the os-
motic pressure of emulsoids are, in general, considerably greater
than the corresponding values obtained with suspensoids. The
osmotic pressures of several typical emulsoids are given in the
following table.

OSMOTIC PRESSURES OF EMULSOIDS.

Sol.

^Concentration
(grams per liter).

Osmotic Pressure
(mm. of mercury).

Egg albumin

12 5

Gelatine

12 5

Starch iodide

Dextrin .

165

It will be observed that the values of the osmotic pressure in the
preceding table are appreciably greater than those given for ferric
hydroxide hydrosols. This is in agreement with the well-estab-
lished fact that emulsoids diffuse more rapidly than suspensoids.
It has been observed, that the value of the osmotic pressure of
gelatine solutions at ordinary temperatures can be increased by
maintaining the solutions at a higher temperature for a short time
and then cooling to the initial temperature. After standing for
several days at the original temperature, however, the osmotic
pressure of the solution returns to its former value. This phenom-
enon would seem to indicate that the osmotic pressure of colloidal
solutions is not completely defined by the two variables, tempera-
ture and concentration. It has been suggested that the degree of
aggregation of the colloid is partially dependent upon the tempera-
ture; the molecular aggregates tending to break up as the tempera-
ture is raised, thus increasing the number of dissolved units and
therefore causing a corresponding increase in the osmotic pressure.

Molecular Weight of Colloids. We have already learned that
the knowledge of the osmotic pressure of a solution enables us to
calculate the molecular weight of the solute, provided the solu-
tion is dilute and obeys the gas laws. As we have seen, other

* Am. Jour. Physiol., 20, 127 (1907).

250 THEORETICAL CHEMISTRY

factors than concentration and temperature determine the osmotic
pressure of colloidal solutions, so that we are not justified in
attempting to calculate the molecular weight of a colloid from the
observed value of the osmotic pressure of its solution. Values for
the molecular weight of colloids calculated from their effect on the
vapor pressure, the boiling-point, and the freezing-point of the
solvent are also untrustworthy, since the same factors which
influence the osmotic pressure necessarily affect these related
properties. This becomes evident when we reflect that an osmotic
pressure of 1 mm. of mercury corresponds to a depression of the
freezing-point of about 0°.0001. Owing to the difficulty of ob-
taining absolutely pure emulsoid sols, all determinations of their
freezing-point depressions must be affected with an experimental
error appreciably larger than the observed depression. Hydrosols
of albumin, gelatine, etc., prepared with extreme care by Bruni
and Pappada * failed to produce any detectable depression of the
freezing-point of water.

Electroendosmosis. The movement of a liquid through a
porous diaphragm, due to the passage of an electric current be-
tween two electrodes placed on opposite sides of the diaphragm, is
known as electroendosmosis. This phenomenon, which was first
observed by Reuss in 1807, has since been made the subject of
numerous investigations by Wiedemann,t Quincke t and Perrin,§
the latter having worked out a satisfactory theoretical interpreta-
tion of the phenomenon. If a porous partition be placed in the
horizontal portion of a U-tube and an electrode be inserted in each
arm of the tube, it will be found, on filling the tube with a feebly
conducting liquid and passing a current, that the liquid will com-
mence to rise in one arm of the tube and will continue to rise until
a definite equilibrium is established. For a given difference of
potential between the two electrodes, there will be a definite differ-
ence in the level of the liquid in the two arms of the tube. The
majority of substances acquire a negative electric charge when

* Rend. R. Accad. dei Lincei, (5), 9, 354 (1900).

t Pogg. Ann., 87, 321 (1852).

t Ibid., 113, 513 (1861).

§ Jour. Chim. Phys., 2, 601 (1904).

COLLOIDS 251

immersed in water. The water, under these conditions, becomes
positively charged and will, in consequence, migrate toward the
cathode. On the other hand, certain substances acquire a positive
charge on immersion in water arid in these cases the direction of
migration will obviously be reversed.

It has been found that acids cause negative diaphragms to be-
come less negative and positive diaphragms to become more posi-
tive. The action of alkalies is, as we should expect, the reverse of
that of acids. There is an interesting connection between the
valence of the ions resulting from the dissociation of dissolved
salts, and the difference of potential existing between the liquid
and the diaphragm. When the diaphragm is positively charged,
the difference of potential is found to be conditioned by the
valence of the anion, and when the diaphragm is negatively
charged, the difference of potential is determined by the valence
of the cation.

^ Cataphoresis. When a difference of potential is established
between two electrodes immersed in a suspension of finely-divided
quartz or shellac, the suspended particles move toward the positive
electrode. This phenomenon is called cataphoresis and was first
observed by Linder and Picton.* They showed that when the
terminals of an electric battery are connected to two platinum
electrodes dipping into a colloidal solution of arsenious sulphide,
there is a gradual migration of the colloid to the positive pole. A
similar experiment with a solution of colloidal ferric hydroxide
resulted in the transport of the dissolved colloid to the negative
pole. It follows, therefore, that the particles of colloidal arsenious
sulphide are negatively charged, while those of colloidal ferric
hydroxide carry a positive charge. It has been found that most
colloids ca^ry an electric charge. In the table on page 252, some
typical colloids arc classified according to the character of their
electrification in aqueous solution.

The nature of the charge varies with the dispersion medium used,
colloidal solutions in turpentine, for example, having charges oppo-
site to those in water.

Direct measurements of the velocity with which the particles

* Jour. Chem. Soc., 61, 148 (1892).

252

THEORETICAL CHEMISTRY

move in cataphoresis have been made by Cotton and Mouton.
By observing with a microscope the distance over which a singk

ELECTRICAL CHARGES OF HYDROSOLS.

Electro-positive.

Electro-negative-

Metallic hydroxides
Methyl violet
Methylene blue
Magdala red
Bismarck brown
Haemoglobin

All the metals
Metallic sulphides
Aniline blue
Indigo
Eosine
Starch

particle traveled in a given interval of time under a definite po-
tential gradient, they calculated the average velocity of migration
of a number of suspensoids. The following table gives the velocity
of migration of a few typical suspensoids.

VELOCITY OF MIGRATION OF SUSPENSOIDS. f

Suspensoid.

Average Diameter
of Particles.

Velocity cm. /sec for
Unit Potential
Gradient.

Arsenic trisulphide. . ....
Quartz ....
Gold (colloidal)

50 M
1 MM

<100 MM

22X10-5
30X10-5
40X10"5

Platinum (colloidal)

< 100 MM

30X10"5

Silver (colloidal)
Bismuth (colloidal)

<100MM
< 100 MM

23 6X10~8
11.0X10™8

Lead (colloidal)... . .
Iron (colloidal)
Ferric hydroxide (colloidal) . .

<1(X)MM
<100MM
100 MM

12.0X10-6
19.0X10-5
30.0X10-5

t Freundlich, Kapillarchemie, p. 234.

It will be observed that not only are the velocities of migration
nearly constant, but also that they are apparently independent of
the size and nature of the particles.

The presence of electrolytes, especially acids and bases, exercises
a marked effect upon the electrical behavior of suspensoids.
Owing to the comparative instability of suspensoids, the addition
of electrolytes usually results in the complete precipitation of the

colloid.

* Jour. Chim. Phys., 4, 365 (1906).

COLLOIDS 253

Emulsoids also show the phenomenon of cataphoresis, but their
velocities of migration are appreciably less than the corresponding
velocities of suspensoids, and their behavior in an electric field is
such as to make it appear quite probable that the character of their
electric charge is entirely fortuitous. Furthermore, emulsoids are
much more susceptible to the influence of electrolytes than are
suspensoids.

W. B. Hardy * has found that the direction of migration of
albumin, modified by heating to 100° C., is dependent upon the
reaction of the dispersion medium. A very small quantity of free
base causes the particles of albumin to move toward the positive
electrode, while the addition of an equally small amount of acid
results in a reversal of the direction of migration. Similar reversals
of charge have been observed by Burton f in colloidal solutions of
gold and silver. When small amounts of aluminium sulphate are
added to colloidal solutions of these metals, the charge is gradually
neutralized and eventually the colloidal particles acquire a reversed
charge.

Electrical Conductance of Colloidal Solutions. The electrical
conductance of suspensoids differs so slightly from that of the pure
dispersion medium that it is difficult to decide whether the small
increase in conductance may not be due to the presence of traces
of adsorbed electrolytes. In order to ascertain to what extent the
conductance of suspensoids is dependent upon the presence of
adsorbed electrolytes, Whitney arid Blake t investigated the effect
of successive electrolyses upon the conductance of a gold hydrosoL
If the pure sol is incapable of enhancing the conductance of the
dispersion medium, then as the sol is subjected to successive elec-
trolyses, the conductance should steadily diminish and ultimately
become identical with that of the dispersion medium. Whitney
and Blake found that the conductance converged to a definite
limiting value which was slightly greater than the conductance of
the dispersion medium. From these experiments we seem to be
warranted in concluding that suspensoids conduct the electric
current very feebly.

* Jour. Physiol., 24, 288 (1899).

t Phil. Mag., 12, 472 (1906).

J Jour. Am. Chem. Soc., 26, 1339 (1904).

254 THEORETICAL CHEMISTRY

Emulsoids appear to conduct rather better than suspensoids.
Whitney and Blake measured the conductance of silicic acid and
gelatine sols and found the specific conductance of the former to be
100 X 10~6 reciprocal ohms and that of the latter to be 68 X 10~6
reciprocal ohms. On the other hand, Pauli* found that an albumin
sol which had been prepared with extreme care was virtually a
non-conductor. It should be remembered, however, that the
albumins are closely related to the simple ammo-acids which are
known to be exceedingly poor conductors.

Precipitation of Colloids by Electrolytes. One of the most
important and interesting divisions of the chemistry of colloids is
that which treats of the precipitation of suspensoids and emulsoids
by electrolytes. In general, it may be said that the precipitation
of colloids by electrolytes is an irreversible process. Colloidal
solutions are more or less unstable systems irrespective of the
methods employed in their preparation, and the addition of a small
amount of an electrolyte is usually found to be sufficient to cause
the sol immediately to become opalescent, and ultimately to pre-
cipitate, leaving the dispersion medium perfectly clear and free
from the disperse phase. Some exceptions to this general state-
ment as to the behavior of colloids are known. For example,
Whitney and Blake f found that precipitated gold could be caused
to return to the colloidal state by treatment with ammonia, while
Linder and Picton J discovered that a ferric hydroxide hydrosol,
which had been precipitated with sodium chloride, could be re-
stored to the colloidal condition by simply removing the electrolyte
with water. The sedimentation of suspensions, such as kaolin in
water, is also promoted by the addition of electrolytes.

On the other hand, the addition of some non-electrolytes fre-
quently causes an increase in the stability of a suspensoid.

Precipitation of Suspensoids. The phenomenon of the pre-
cipitation of suspensoids has been carefully investigated by
Freundlich.§ He has found that an amount of electrolyte which

* Beitrag, Chem. Phys. Path., 7, 531 (1906).

t Jour. Am. Chem. Soc., 26, 1341 (1904).

J Jour. Chem. Soc., 61, 114 (1892); 87, 1924 (1905).

§ Zeit. phys, Chem., 44, 131 (1903).

COLLOIDS

255

is incapable of bringing about an instantaneous precipitation, may
become effective after an interval of time. He has also shown
that the total quantity of electrolyte required to precipitate a
suspensoid completely depends upon whether the electrolyte is
added all at one time or in successive portions. In order to com-
pare the precipitating action of various electrolytes, Freundlich
proposed the following procedure, which prevents the possibility
of irregularities due to the time factor: — To 20 cc. of a solution
of a suspensoid, 2 cc. of the solution of the electrolyte are added,
the resulting solution being shaken vigorously ; the mixture is then
set aside for two hours, after which a small portion is filtered off,
and the filtrate is examined for the suspensoid. In the following
table some of the results obtained by Freundlich with colloidal solu-
tions of ferric hydroxide are given. The data represent the mini-
mum concentration for each electrolyte which produced precipita-
tion in two hours.

It will be seen that very small amounts of the electrolytes are
required to precipitate the suspensoid, and further, that the pre-
cipitating power of an electrolyte is dependent upon the charge of
the negative ion. The greater the charge, the smaller is the
quantity of electrolyte required to produce precipitation.

PRECIPITATING ACTION OF ELECTROLYTES ON FERRIC

HYDROXIDE HYDROSOL.
(16 milli-mols Fe(OH)2 per liter.)

Electrolyte.

Concentration
(milli-inols per
hter).

NaCl

9 25

KC1

9 03

BaCl2

9 64

KNO3

11.9

KBr

12 5

Ba(N03)2

14 0

16 2

HC1

400 ca.

Ba(OH)2

0 42

H3(SO)4

0 204

MgS04

0 217

K2Cr2O7

0.194

H*SO4

0.5ca.

256

THEORETICAL CHEMISTRY

The significance of the relation between ionic charge and pre-
cipitating power was first pointed out by Hardy,* who formulated
the following rule: — The precipitation of a colloidal solution is de-
termined by that ion of an added electrolyte which has an electric charge
opposite in sign to that of the colloidal particles.

It has already been pointed out that colloidal particles of arse-
nious sulphide are negatively charged, hence, according to Hardy's
rule, the positive ions of the added electrolyte will condition the
precipitation of the suspensoid. The experiments of Freundlich
confirm this prediction, as is shown by the following table:

PRECIPITATING ACTION OF ELECTROLYTES ON ARSENIOUS
SULPHIDE HYDROSOL.

(7.54 milli-mols As2Ss per liter.)

Electrolyte.

Concentration
(milh-mols per
liter).

KC1

49 5

KNO3

50 0

KC2H3O2

110 0

NaCl

51 0

LiCl

58.4

MgCl2

0.717

MgS04

0 810

CaCl2

0.649

SrCh

0 635

BaCi2

0 691

Ba(N03)2

0 687

ZnCl2

0 685

AlClr

0 093

A1(N03)3

0095

Precipitation and Valence* An examination of the preceding
tables reveals the fact that although the ionic concentration neces-
sary to bring about precipitation, in accordance with Hardy's rule,
decreases with increasing valence, the diminution in concentration
is not, as we might expect, inversely proportional to the valence of
the precipitating ion. The absence of any simple quantitative
relation between the valence of an ion and its precipitating con-
* Zeit. phys. Chem., 33, 385 (1900),

COLLOIDS 257

centration is undoubtedly due to the influence of several potent
factors, such as adsorption and the protective action of ions whose
electric charge is of the same sign as that of the colloidal substance.

Precipitation of Emulsoids. The action of electrolytes on
emulsoids is much more obscure than the action of electrolytes on
guspensoids. Nothing approaching a generalization similar to
Hardy's rule for the precipitation of suspensoids has been found to
apply to the precipitation phenomena manifested by emulsoids.
Owing to the fact that emulsoids are liquids, and in consequence
of their greater degree of dispersity, it has been suggested that
emulsoids probably resemble true solutions more closely than sus-
pensoids. In fact there is reason for assuming that a portion of
the colloid is actually dissolved in the dispersion medium. This
may account for the fact that the precipitation of emulsoids is
sometimes reversible and sometimes irreversible. It is to be re-
gretted that, up to the present time, so many of the investigations
on emulsoids have been carried out with materials of questionable
purity and of insufficient uniformity.

The addition of salts to gelatine generally causes irreversible
precipitation, provided the concentration of the salt is not too low.

The precipitation of albumin by some salts is reversible while
by others it is irreversible. Those transformations which are
initially reversible gradually become irreversible on standing.
Although relatively concentrated solutions of salts of the alkalies
and alkaline earths are required to precipitate albumin, very dilute
solutions of the salts of the heavy metals are found to be sufficient
to bring about complete precipitation.

Action of Heat on Emulsoids. When an albumin hydrosol is
gradually heated, a temperature is ultimately reached at which
coagulation occurs. The exact nature of this transformation is
not understood, but it is believed to be largely chemical. This
belief is based upon the fact that the reaction of the natural albu-
mins toward litmus is altered by heating. Slight acidity of the sol
is essential to complete coagulation, while an excess or a de-
ficiency of acids causes a portion of the albumin to remain in the
sol. The presence of various salts has been found to exert a
marked influence on the temperature of coagulation of albumin.

258 THEORETICAL CHEMISTRY

The coagulation temperature is invariably raised at first, attaining
a constant value in some cases, while in others it decreases after
reaching a maximum temperature. It is especially interesting to
note that the anions of the salts follow the usual lytropic sequence.
The effect of heat on a gelatine sol is very different from its
effect on an albumin sol. If a fairly concentrated gelatine sol is
heated and then permitted to cool, it sets into a jelly which is not
reconverted into a sol when the temperature is again raised. Fur-
thermore, the change does not take place at a definite temperature.
In studying the phenomenon of gelation, either the temperature
or the time of gelation may be determined. The melting-point of
pure gelatine ranges from 26° to 29° while the solidifying tempera-
ture lies between 25° and 18°. The melting and solidifying tem-
perature of gelatine sols vary with the concentration ; a 5 per cent
sol melts at 26°. 1 and solidifies at 17°.8, while a 15 per cent sol
melts at 29°.4 and solidifies at 25°.5. The temperature of the gel-
sol transformation is affected by the addition of salts, some tending
to raise the temperature of gelation and others to lower it. The
order of the anions arranged according to their influence on the
gel-sol transformation is as follows: —

Raising temperature: S04 > Citrate > Tartrate > Acetate (H20).
Lowering temperature: (H2O) 01 < C103 < NO3 < Br < I.

The same lytropic order was found by Schroeder * in an investi-
gation of the viscosity of gelatine sols.

It is noteworthy that the influence of salts on the temperature
of gelation of agar-agar and other similar substances is analogous
to their effect on gelatine, the same lytropic sequence being main-
tained.

Protective Colloids. The precipitating action of electrolytes
on suspensoids may be inhibited by adding to the solution of the
suspensoid a reversible colloid. The protective action of a re-
versible colloid is not due, as might be supposed, to the increased
viscosity of the medium and the resultant resistance to sedimenta-
tion, since amounts of a reversible colloid, too minute to produce
any appreciable increase in the viscosity of the medium, can pre-

* Zeit. phys. Chem., 45, 75 (1903).

COLLOIDS

259

vent precipitation. Thus, Bechhold * has shown that while a
mixture of 1 cc. of a suspension of mastic and 1 cc. of a 0.1 molar
solution of MgSO4 diluted to 3 cc. with water, is completely pre-
cipitated in 15 minutes, no precipitation will occur within 24 hours,
if two drops of a 1 per cent solution of gelatine be added before
diluting to 3 cc.

Gum arabic and ox-blood serum exert a similar protective action
when added to a suspension of mastic. The protective power of
reversible colloids differs widely and Zsigmondy f has attempted
to make this the basis of a method of classification of colloidal sub-
stances. A red solution of colloidal gold becomes blue on the
addition of a small amount of sodium chloride, owing to the in-
crease in the size of the colloidal particles. Various colloidal sub-
stances when added to a red colored gold sol protect the colloidal
particles from precipitation by a solution of sodium chloride, no
change in color following the addition of the electrolyte. A definite
amount of each colloidal substance is required to prevent the
change from red to blue in the color of the gold sol. In employing
this color change as a means of differentiating colloidal substances,
Zsigmondy introduced the "gold number," which may be defined
as the weight in milligrams of a colloidal substance which is just
insufficient to prevent the change from red to blue in 10 cc. of a
gold sol after the addition of 1 cc. of a 10 per cent solution of
sodium chloride. The following table gives the gold numbers of a

few colloids.

GOLD NUMBERS OF COLLOIDS.

Colloid.

Gold Number.

Gelatine
Casein (in ammonia) ....
Egg-albumin

0 005-0 01
0 01
0 15-0 25

Gum arabic

0 15-0 25; 0 5-4

Dextrin

6-12; 10-20

Starch, wheat

4-6 (about)

Starch, potato

25 (about)

Sodium stearate

10 (at 60°);0 01 (at

100°)

Sodium oleate

0.4-1

Cane sugar

Urea

* Zeit. phys. Chem., 48, 408 (1904).
t Zeit. analyt. Chem., 40, 697 (1901).

260

THEORETICAL CHEMISTRY

The gold number has proven useful in differentiating the various
kinds of albumin, as is shown in the following table.

GOLD NUMBERS OF ALBUMINS.

Albumin.

Gold Number.

Egg white (fresh)

0 08

Globulin . .

0 02-0 05

Ovomucoid .

0 04-0 08

Albumin (Merck)
Albumin (cryst.) ....
Albumin (alkaline)

0 1-0 3
2-8
0 006-0 04

The addition of alkali to any one of the first five albumins of the
above table reduces the gold number to that of alkaline albumin.
Sulphide sols may be protected as well as metallic sols, and further-
more the ability to exert protective action is not confined to organic
colloids alone.

In general, it may be said that when a suspensoid sol is mixed
with an emulsoid sol in the proper proportions, the suspensoid sol
acquires most of the characteristic properties of the protecting
colloid. The masking of the properties of a suspensoid sol by a
protecting colloid is probably to be ascribed to the formation of a
thin film of adsorbed emulsoid over the suspensoid.

Reciprocal Precipitation. A further deduction of the elec-
trical theory of precipitation is, that when two oppositely-charged
colloids are mixed, they should precipitate each other, and the
resulting precipitate should contain both colloids. Experiments
carried out by Biltz * have confirmed these predictions. He
showed that when a solution of a positively-charged colloid is
added to a solution of a negatively-charged colloid, precipitation
occurs, unless the quantity of the added colloid is either relatively
very large or very small. He also showed that when two colloids
of the same electrical sign are mixed no precipitation occurs. Just
as the amount of precipitation caused by the addition of an elec-
trolyte to a sol is conditioned by the rate at which the electrolyte
is added, so also the precipitation of one colloid by another is de-

* Berichte, 37, 1095 (1904).

COLLOIDS

261

pendent upon the manner in which the two sols are mixed. The
extent to which one sol is precipitated by another sol of opposite
sign is largely determined by the amount of one that is added to a
definite amount of the other. This is clearly shown by the data
of the following table which give the results obtained by Biltz on
adding ferric hydroxide sol to 2 cc. of an antimony trisulphide sol
containing 2.8 mg. per cc.

PRECIPITATION OF COLLOIDAL ANTIMONY TRISULPHIDE
BY COLLOIDAL FERRIC HYDROXIDE.

Fe203(m*.).

Immediate Result.

Result After One Hour.

0.8
3 2
4.8
6.4
8.0
12 8
20.8

Cloudy
Small flakes
Flakes
Complete precipitation
Slow precipitation
Cloudy
Cloudy

Almost homogeneous
Unchanged
Yellow liquid
Complete precipitation
Complete precipitation
Slight precipitation
Homogeneous

It will be seen that the addition of a small amount of ferric
hydroxide produces hardly any precipitation. With the addition
of larger amounts of ferric hydroxide, the amount of precipitation
increases until finally complete precipitation is attained. The
addition of larger quantities of ferric hydroxide produces either
little or no precipitation. It has been found that at the concen-
tration which just produces complete precipitation, the electrical
charges of the two sols are equivalent. When the amount of ferric
hydroxide exceeds that required for complete precipitation, it is
more than probable that the particles of colloidal antimony tri-
sulphide are completely enveloped by the particles of ferric hy-
droxide and thereby rendered inactive.

When we come to study the action of one emulsoid on another,
we find, as might be expected from the general behavior of emul-
soids toward electrolytes, that the phenomena are more complex
and very much less well-defined than with suspensoids. Although
mutual precipitation does take place with emulsoids, the close
resemblance between emulsoids and true solutions renders the
phenomenon more or less indistinct.

262 THEORETICAL CHEMISTRY

When a suspensoid sol is added to an emulsoid sol having an
opposite electric charge, precipitation may or may not occur
according to the relative amounts of the two colloids in the mix-
ture. When the two colloids are present in electrically equivalent
quantities precipitation occurs, otherwise one colloid exerts a
protective action on the other.

Characteristics of Gels. Gels are generally obtained by cooling
or evaporating emulsoid sols and, since the latter are known to be
two-phase liquid systems, it is natural to infer that gels may also
be two-phase systems. According to this conception, the only
difference between an emulsoid sol and a gel is, that in the latter
the concentration of at least one of the phases is greatly increased
and thereby imparts greater viscosity and rigidity to the system.
It has been suggested that the more concentrated of the two phases
forms the walls of an assemblage of cells within which the more
dilute phase is enclosed. The view that gels possess a distinct
cellular structure is fully confirmed by microscopic examination.

The extreme sensitiveness of gels changes in temperature and
to the presence of extraneous substances renders their investi-
gation exceedingly difficult. Notwithstanding the experimental
difficulties involved in the study of gels, sufficient knowledge has
been gained of their properties to make a brief account of these
necessary in any treatment of the subject of colloids.

Physical Properties of Gels. The process of gel formation
from a dry gelatinous colloid and water invariably involves con-
traction. This statement should not be confused with the fact
that a gel on immersion in water undergoes appreciable increase in
volume.

Gels have been shown by Barus * to be considerably more com-
pressible than solids. The compressibility increases as the tem-
perature is raised until, when the gel is transformed into the sol,
the compressibility becomes equal to that of pure water. The
temperature of some gels, such as rubber and gelatine, is lowered
by compression and raised by tension.

The thermal expansion of gels is nearly identical with that of
the more fluid component of the gel.

* Am. Jour. Sci., 6, 285 (1898).

COLLOIDS 263

The rate at which pure substances diffuse in gels differs only
slightly from the rate of diffusion in pure water, provided the con-
centration of the gel is not too great. The slight resistance offered
by gels to diffusion of dissolved substances may be regarded as
further evidence in favor of their cellular structure.

The modulus of elasticity of a gel, cast in a cylindrical mold, is
given by the formula

E - Pl

where P is the tension which produces the increase in length AH in
a cylinder whose length is I and whose radius is r. It has been
found that the modulus of elasticity in gelatine gels increases as the
square of the concentration of the gel. The time of recovery, after
releasing the tension, increases as the concentration of the gel
increases.

The shearing modulus for a gel is given by the formula

*•

"•-27T+7)'

where /i denotes the ratio of the relative contraction of the diameter
to the relative change in length. The viscosity of a gel may be
calculated from the shearing modulus by means of the equation

11 = E8T}

where r denotes the time of recovery. Since both E8 and r increase
with the concentration of the gel it is apparent that the viscosity
of the gel must also increase with the concentration.

As is well known, when glass is subjected to pressure or is un-
equally strained, it exhibits the phenomenon of double refraction.
Since glass bears some resemblance to gels in being a highly viscous,
supercooled liquid, it might reasonably be inferred that gels should
also show double refraction. Experiments with collodion and
gelatine have shown that these substances, when subjected to
pressure, behave similarly to glass.

Hydration and Dehydration of Gels. The complementary
processes of hydration and dehydration of gels are extremely inter-

264 THEORETICAL CHEMISTRY

esting. Following Freundlich * we will consider the subject very
briefly under the two following heads: (a) Non-elastic Gels and
(b) Elastic Gels.

(a) Non-elastic Gels. When freshly prepared aluminium or fer-
ric hydroxide gels were placed in a desiccator, it was found by
van Bemmelen f that the rate at which these substances lost water
was continuous. Furthermore, on removing the dried gels from
the desiccator it was found that the process of recovery of moisture
was also continuous. It will be shown in a subsequent chapter
(p. 333) that a definite hydrate in the presence of its products of
dissociation, possesses a constant vapor pressure so long as any of
that particular hydrate is present. The fact that the vapor
pressure of the gels investigated by van Bemmelen did not remain
constant but decreased continuously as the water was removed,
proved conclusively that no chemical compounds were formed in
the process of gel hydration.

Another gel which has been made the subject of much careful
investigation is silicic acid. The dehydration curve of silicic acid
is continuous, with the exception of a short portion where an ap-
preciable amount of water is lost without much of any change in
the vapor pressure. This portion of the curve corresponds to a
marked change in the appearance of the gel. The gel which had
hitherto been clear and transparent became opalescent soon after
the vapor pressure had attained a temporarily constant value.
The opalescence gradually permeated the entire mass, until the
gel acquired a yellow color by transmitted light, and a bluish color
by reflected light. These colors suggest a marked increase in the
degree of dispersity of the gel, an inference the correctness of
which subsequent investigation has fully confirmed.

The curve of hydration, while resembling the curve of dehydra-
tion in many respects, departs quite widely from it in others. The
change in dispersity, as indicated by the appearance in reverse
order of the color and opalescent phenomena mentioned above,
also was manifest.

* Kapillarchemie, p. 486.

t Zeit. anorg. Chemie, 5, 466 (1894); 13, 233 (1897); 18, 14, 98 (1898); 30,

265 (1902).

COLLOIDS 265

(b) Elastic Gels. The absence in elastic gels of a horizontal
portion in the dehydration curves, and the fact that although
elastic gels may have become saturated with water vapor, they
still retain the power of taking up large amounts of liquid water
when immersed in that medium, constitute the chief differences
between elastic and non-elastic gels.

The amount of water which can be taken up by an elastic gel is
exceedingly large. Thus, on exposing a plate of gelatine weighing
0.904 gram for eight days in an atmosphere saturated with water
vapor, Schroeder * found that it had taken up 0.37 gram of water.
On exposing for a longer period of time under the same conditions,
he found no further gain in weight, and on removing the gelatine
plate from the moist atmosphere and placing it in a desiccator, the
plate slowly gave up the absorbed moisture and regained its origi-
nal weight. On the other hand, when the plate, after having
absorbed the maximum weight of moisture from the air, was im-
mersed in water, it was found to increase in weight very rap-
idly. Thus, on immersing the above plate which weighed 1.274
grams when saturated in moist air, and allowing it to remain
in water for one hour, it was found to have taken up 5.63
grams of water. After an immersion of twenty-four hours, the
plate was found to have taken up the maximum weight of water
it was capable of absorbing at that temperature. On remov-
ing the plate, it was found to part with the absorbed water
very readily, even in moist air, the greater part of the absorbed
water being so loosely held that the vapor pressure of the gel
remained the same as that of pure water at the temperature of
the experiment.

It is impossible to measure directly the pressure produced by gels
when they take up water, but some idea of the magnitude of these
pressures may be obtained by coating a glass plate with gelatine,
which has imbibed the maximum amount of water, and observing
the degree to which the glass plate is bent by the drying gelatine
film. Frequently the elastic limit of the glass is exceeded and
the plate breaks under the stress produced by the dehydration of

the gel.

* Zeit. phys. Chem., 45, 75 (1903).

266

THEORETICAL CHEMISTRY

Velocity of Imbibition. The rate at which gels imbibe water
has been studied by Hofmeister.* The velocity of imbibition of
water by thin plates of gelatine and agar-agar was measured by
removing the plates at definite intervals and determining their
increase in weight. Owing to the time consumed in making the
weighings, the time intervals are affected by an appreciable error.
The following table gives the data of a single experiment with
gelatine.

VELOCITY OF IMBIBITION IN GELATINE.

(Thickness of plate 0.5 mm.)

Time (mm )

Water Imbibed
(grains)

3 08

0 090

3.88

0 084

4 26

0 084

4 58

0 064

4 67

0 075

4 96

If the weight of water imbibed in i minutes is Wt, and wx is the
maximum weight of water which a gel can take up under the con-
ditions, then the velocity of imbibition should be given by the
equation

dw , , ,

-fi = k(w»-wt),

which on integration becomes

"If — __j[{-»p* dO. t

t (WQO — Wt)

The figures in the third column of the preceding table were cal-
culated by means of this equation and, although the values are not
strictly constant, the variation is no greater than might be expected
where the experimental error is so large.

Heat of Imbibition. The process of imbibition is accom-
panied by an evolution of heat. Quantitative measurements

* Arch. exp. Pathol. u. Pharmakol., 27, 395 (1890).

COLLOIDS

267

of the heat evolved when gels take up water have been made
by Wiedemann and Liideking * and also by Rodewald.f The
following table gives Rodewald's data on the heat of imbibition
of starch.

HEAT OF IMBIBITION OF STARCH.
(Weight of dry starch 100 grams.)

Per Cenjl, Water.

Heat in Calories per
Gram of Starch.

0 23

28.11

2 39

22 60

6 27

15 17

11 65

8 43

15 68

5 21

19 52

2 91

It will be observed that the greatest development of heat ac-
companies the initial stages of imbibition where very small
amounts of water are taken up. This is what we might expect
when we remember that it is the last remaining portion of water
which is most difficult to remove from a gel, and that it is only
through the application of heat that its complete removal can be
effected.

Imbibition in Solutions. When a gel is immersed in a saline
solution, the salt distributes itself between the solvent and the
gel. The rate of imbibition is found to vary greatly according
to the salt which is present in the solution. Thus, the velocity
of imbibition has been found to be accelerated by the presence
of the chlorides of ammonium, sodium and potassium and by
the nitrate and bromide of sodium; on the other hand, the pres-
ence of the nitrate, sulphate and tartrate of sodium retard im-
bibition.

The effect of acids and bases on imbibition appears to be similar
to the influence of salts.

Adsorption. The change in concentration which occurs at
the boundary between two heterogeneous phases is termed adsorp-

* Wied. Ann., 25, 145 (1885).

t Zeit. phys. Chem., 24, 206 (1897).

268 THEORETICAL CHEMISTRY

twn. At the surface of a solid surrounded by a gas or vapor, the
phenomenon is generally known as gaseous adsorption, since any
difference which may occur in the concentration of the solid phase
is much too small to be detected. At the boundary between
liquid and gaseous phases, the concentration of each phase un-
doubtedly undergoes alteration. In the case of the boundary be-
tween solid and liquid phases, the only apparent inequality in
concentration occurs on the liquid side of the boundary, notwith-
standing the fact that the adsorbed substance is quite commonly
regarded as being bound to the surface of the solid phase. The
cause of this erroneous conception is, that the extremely thin layer
of liquid in which the alteration in concentration actually occurs,
is the layer which wets the surface of the solid and hence is the
layer which adheres to the solid when it is removed from the
liquid.

The retention of gases by charcoal is a typical example of gaseous
adsorption, while the removal of coloring matter by charcoal in the
purification of various organic substances may be cited as an
example of adsorption of a liquid by a solid.

If the adsorbed substance increases in concentration in the
vicinity of the boundary, the adsorption is said to be positive; if it
decreases, the adsorption is said to be negative,

Adsorption of Gases. In gases, adsorption-equilibrium is
attained with remarkable rapidity. Thus, if a gas is admitted
into a vessel containing some freshly prepared cocoanut charcoal,
the pressure will fall immediately to a value which corresponds to
the removal of the entire adsorbed volume of gas.

The concentration of adsorbed gas on the surface of a solid, when
equilibrium is attained, has been shown to be approximately
1 X 10~7 gram per square centimeter. This value is of the same
order of magnitude as the strength of the limiting capillary layer
of a liquid and, therefore, lends support to the suggestion put for-
ward some years ago by Faraday that an adsorbed film of gas may
be present in the liquid state.

The amount of gas adsorbed by a solid increases with the pres-
sure and diminishes with increasing temperature. The following
empirical equation, expressing the relation between the amount of

COLLOIDS 269

gas adsorbed by a solid and the pressure, has been proposed by
Freundlich *

— = apn,

m '

where x is the total mass of gas adsorbed on a surface of m sq.
cm. under a pressure p, and where a and n are constants. This
equation, known as the adsorption isotherm, has been found to hold
quite generally.

Adsorption in Solutions. The adsorption phenomena occur-
ring at the surface of contact of a solid with a solution are similar
to the phenomena which have just been discussed. Because of the
frequency of its occurrence in many of the more common operations
of both laboratory and factory, the subject of adsorption in solu-
tions deserves fuller treatment.

The general characteristics of adsorption in solutions may be
briefly summarized as follows: —

(1) Adsorption in solutions is generally positive, i.e., on shaking
a solution with a finely-divided adsorbent, the volume concentra-
tion of the solution will diminish.

(2) The amount of positive adsorption may be sufficient to re-
move almost all of the solute from a solution, especially if the solu-
tion is dilute. On the other hand, negative adsorption is always
very small, and frequently is immeasurable.

(3) Adsorption is directly proportional to the so-called " specific
surface," the latter term being defined as the ratio of the total sur-
face of the adsorbent to its volume.

(4) On shaking a definite weight of an adsorbent with a given
volume of solution of known concentration, a definite equilibrium
will be established. If the solution is then diluted with a known
amount of solvent, the adsorption will decrease until it acquires the
same value which it would have attained, had the same weight of
adsorbent been introduced directly into the more dilute solution.
For example, if 1 gram of charcoal is agitated with 100 cc. of a
0.0688 molar solution of acetic acid for 20 hours, adsorption is
found to reduce the original concentration of the acid to 0.0678

molar.

* Zeit. phys. Chem., 57, 385 (1906).

270 THEORETICAL CHEMISTRY

In a second experiment, if 1 gram of charcoal is shaken for the
same period of time with 50 cc. of 2 X 0.0688 = 0.1376 molar
acetic acid, and then, after adding 50 cc. of water, the shaking is
continued for an additional period of 3 hours, the final concentration
of the acid will be found to be the same as in the first experiment.

(5) It is impossible to determine the specific surface of an ad-
sorbent directly owing to its porosity. However, according to (3)
adsorption is directly proportional to the specific surface and there-
fore the weights of different adsorbents which produce the same
amount of adsorption may be assumed to possess equal specific
surfaces.

(6) Adsorption in solution is largely dependent upon the surface
tension of the solvent. In solutions of the same substance in
different solvents, the greatest adsorption occurs in that solution
whose solvent possesses the highest surface tension.

(7) The order of efficiency of adsorption is not only independent
of the nature of the solvent but also of the nature of the adsorbed
substance.

The Adsorption Isotherm. The empirical equation of Freund-
lich for gaseous adsorption has been found to apply equally well to
adsorption equilibria in solutions. The equation may be written
as follows:

— = acn.
m '

where x is the weight of substance adsorbed by a weight m of ad-
sorbent from a solution whose volume-concentration at equilibrium
is c, and where, as before, ex. and n are constants. The constant n
varies in different cases from n = 2 to n = 10; within these limits
the value of n is independent of the temperature and also of the
natures of the adsorbed substances and the adsorbent. Although
the value of the constant a varies over a wide range, the ratio of its
values for two adsorbents in different solutions is practically con-
stant.

The following table contains the data given by Freundlich * on
the adsorption of acetic acid by charcoal.

* Kapillarchemie, p. 147.

COLLOIDS

271

ADSORPTION OF AQUEOUS ACETIC ACID BY CHARCOAL.
t = 25°; « = 2.606; 1/n = 0.425.

Concentration
(mols per liter).

x/m (obs.).

x/m (calc.).

0 0181

0 467

0 474

0 0309

0 624

0 596

0 0616

0.801

0 798

0 1259

1 11

1.08

0 2677

1 55

1.49

0.4711

2.04

1 89

0.8817

2 48

2.47

2.785

3 76

4 01

The validity of the adsorption isotherm is best tested graphically,
by plotting the logarithms of the experimentally determined values
of x/m against the logarithms of the corresponding concentrations.
If the equation holds, a straight line should be obtained. The
curves shown in Fig. 69 represent x/m as a function of c, and
log x/m as a function of log c: it will be observed that the loga-
rithmic plot is practically rectilinear.

log C

c
Fig. 69.

Surface Energy of Colloids. In almost all colloidal solutions
there exists a difference of potential between the particles of the col-
loid and the surrounding medium. The importance of this factor

272 THEORETICAL CHEMISTRY

in interpreting the behavior of colloids has already been empha-
sized. Another factor of equal importance in connection with
colloidal phenomena, is that which depends upon the enormous
surface of contact between the colloid and the surrounding medium
There is an abundance of evidence showing that a colloidal solution
is non-homogeneous, or in other words, that it is essentially a sus-
pension of finely-divided particles in a fluid medium. An immense
increase in superficial area results from the division and sub-
division of matter. To bring about this comminution requires a
large expenditure of energy. In a colloidal solution this energy is
stored up in the colloidal particles in the form of surface energy,
which may be defined as the product of surface area and surface
tension.

For example, suppose 1 cc. of a substance to be reduced to
cubical particles measuring 0.1 M on each edge, and let the particle
be suspended in water at 17° C, The total energy involved can
be calculated as follows: — The volume of a single particle is
0.1 /i3 or 1 X 10~15 cc.; hence the total number of particles is
1 X 1015. The surface of a single particle is 6 X (0.1 /z)2, or
6 X 10~10 sq. cm., and the total surface is 6 X 105 sq. cm. The
surface tension of water at 17° C. is 71 dynes; hence the total
surface energy is 71 X 6 X 105 = 4.32 X 107ergs. This enormous
figure shows that where the surface of the disperse phase is highly
developed, as it is in colloidal solutions, the surface energy becomes
a very important factor in determining the behavior of the system.
This is especially the case when the degree of aggregation of the
colloidal particles is changed, since a relatively small change in the
amount of aggregation may involve a great change in the surface
exposed and a corresponding change in the surface energy. A very
close connection exists between the electrical and surface factors in
a colloidal solution.

i Surface Concentration. It has been pointed out in an earlier
chapter (p. 144) that as the result of unbalanced molecular attrac-
tion, the surface of a liquid behaves like a tightly stretched mem-
brane. In consequence of this contractile force, or surface tension,
the pressure at the surface of a liquid is greater than the pressure
within the liquid.

COLLOIDS 273

The experiments of Soret (p. 208) and the theoretical deductions
of van't Hofif have shown that when a dilute solution is unequally
heated, the solute distributes itself in accordance with the gas laws,
the solution becoming more concentrated in the cooler portion.
Just as the homogeneity of a dilute solution has been shown to
be disturbed by inequality of temperature, so also inequality of
pressure may be assumed to cause differences in concentration in
the solution. Although direct experimental verification is difficult,
there is abundant evidence for the view that the concentration at
the surface of solution differs from the volume-concentration of
the solution in consequence of the greater pressure in the surface
layer.

The mathematical relation between surface concentration and
surface tension was first deduced by J. Willard Gibbs * in 1876.
The following simplified derivation of this important equation is
due to Ostwald. Let s be the surface of a solution whose surface
tension is 7, and let it be assumed that the surface contains 1 mol
of the solute. If a very small portion of the solute enters the sur-
face layer from the solution, thereby causing a diminution dy in
the surface tension, the corresponding change in energy will be
s dy. But this gain in energy must be equivalent to the osmotic
work involved in effecting the removal of the same weight of solute
from the solution. Let v be the volume of solution containing
unit weight of solute, and let dp be the difference in the osmotic
pressures of the solution before and after its removal; the osmotic
work will be —vdp. Since the gain in surface energy and the
osmotic work are equal, we have

sdy = —vdp.

The solutions being dilute, we may assume that the gas laws hold,
and since v = RT/p, we may write

RT ,
~—dp,

^
OT dp sp

* Trans. Conn. Acad., Vol. Ill, 439 (1876).

274 THEORETICAL CHEMISTRY

Since pressure is directly proportional to concentration, the pre-
ceding equation becomes

« _.

dc sc

But s has already been defined as the surface which contains 1 mol
of solute in excess, from which it follows that the excess of solute
in unit surface is I/a. Writing u = I/a, we have

„- -JL*L

U~~ RT dc
which is the equation of Gibbs.

From this equation it is evident that if the surface tension, 7,
increases with the concentration, then u is negative and the surface
concentration is less than the concentration of the bulk of the
solution. This is clearly negative adsorption. On the other
hand, if 7 decreases as the concentration increases, u is positive and
the surface concentration is greater than the concentration of the
bulk of the solution, or the adsorption is positive. Finally, if the
surface tension is independent of the concentration, then the con-
centration of the solute in both the surface layer and the bulk of
the solution will be the same.

Preparation of Colloidal Solutions. Since 1861, when Graham
published his first paper on colloids, numerous investigators have
devised methods for the preparation of colloidal solutions. Within
recent years our knowledge of this class of solutions has been
greatly increased, many crystalloidal substances having been
obtained in the colloidal condition. As a result of these investi-
gations, we no longer speak of crystalloidal and colloidal matter,
but use the terms crystalloid and colloid to distinguish two dif-
ferent states. In fact it is now recognized that it is simply a
matter of overcoming certain experimental difficulties, before it
will be possible to obtain all forms of matter in the colloidal
state. The scope of this book forbids a detailed account of the
various methods which have been devised for the preparation of
colloidal solutions.* We must content ourselves with a general
classification of these methods into two groups as follows: —

* See "Die Methoden zur Herstellung Kolloider Losungen anorganischer
Stoffe," by Theodore Svedberg, Dresden, 1909.

COLLOIDS 275

(1) Crystallization Methods, and (2) Solution Methods. These
two divisions are sufficiently comprehensive to include all of the
known methods for the preparation of colloidal solutions, with the
possible exception of the electrical methods which may be con-
sidered as forming a separate group.

Crystallization Methods. The crystallization methods include
the following subdivisions: —

(1) Methods involving cooling of a liquid or solution.
Example: — On cooling an alcoholic solution of sulphur in liquid

air, a transparent, highly dispersed, solid sol is obtained.

(2) Methods involving change of medium.

Example: — On gradually adding a solution of mastic in alcohol
to a large volume of water, the mastic is precipitated in a finely
divided condition and a colloidal mastic hydrosol results.

(3) Reduction methods.

Example: — On adding a cold, dilute solution of hydrazine
hydrate to a dilute, neutral solution of auric chloride, a dark blue
gold sol is obtained.

In addition to hydrazine, numerous other reducing agents may
be employed, such as phosphorus, carbon monoxide, hydrogen,
acetylene, formaldehyde, acrolein, various carbohydrates, hy-
droxylamine, phenylhydrazine, and metallic ions.

(4) Oxidation methods.

Example: — On oxidizing a solution of hydrogen sulphide by air
or sulphur dioxide, a colloidal solution of sulphur is obtained.

(5) Hydrolysis methods.

Example: — When a solution of ferric chloride is slowly added to
a large volume of boiling water, the salt undergoes hydrolysis, and
on cooling the dilute solution, a reddish-brown ferric hydroxide
sol is obtained.

(6) Methods involving metathesis.

Example: — A colloidal solution of silver may be prepared by
adding a few drops of a dilute solution of sodium chloride to a dilute
solution of silver nitrate, provided the resulting solution of sodium
nitrate is below the precipitating concentration.

276 THEORETICAL CHEMISTRY

Solution Methods. Under this heading are to be grouped the
so-called "peptization" methods. The term, peptization, was intro-
duced by Graham to express the transformation of a gel into a sol.
To-day we understand a peptizer to be a substance which, if
sufficiently concentrated, is capable of effecting the solution of a
solid which is insoluble in its dispersion medium. A typical
example of peptization is afforded by silver chloride which forms a
sol on prolonged digestion with a solution which contains either
Ag'or Cl'. It is apparent that the rate of peptization can be con-
trolled by dilution of the peptizer, and that when the sol stage is
attained, the peptizer may be readily removed by dialysis.

Numerous reactions are known in which the conversion of an
insoluble precipitate into a sol can only be effected through the
removal of the excess of electrolyte by prolonged washing or dial-
ysis. A familiar example of this type of peptization is furnished
by the tendency of many precipitates to run through the filter
after too prolonged washing with water.

Electrical Methods. These methods of preparing colloidal solu-
tions depend upon the dispersive action of a powerful electric
discharge upon compact metals. In 1897 Bredig * discovered,
while studying the action of the electric current on different
liquids, that if an arc be established between two metallic wires
immersed in a liquid, minute particles of metal are torn off from
the negative terminal and remain suspended in the liquid indefi-
nitely. In order to prepare a colloidal solution by the method of
electrical dispersion, Bredig recommends that a direct current
arc be established between wires of the metal of which a colloidal
solution is desired, the ends of the wires being submerged in water
in a well-cooled vessel, as shown in Fig. 70. The current employed
ranges in strength from 5 to 10 amperes, and the voltage lies be-
tween 30 and 110 volts. A rheostat and an ammeter are included
in the circuit.

The wires are brought in contact for an instant in order to
establish the arc, after which they are separated about 2 nun.
During the gentle hissing of the arc, clouds of colloidal metal are*
projected out into the water from the negative wire, a portion of

* Zeit. Elektrochem., 4, 514 (1897); Zeit. phys. Chem., 31, 258 (1899).

COLLOIDS 277

the metal torn off being distributed through the water as a coarse
suspension. The size of the particles disrupted from the negative
terminal is dependent upon the strength of the current, a current

Ammeter

Fig, 70.

of 10 amperes producing a greater proportion of colloidal metal
than a current of 5 amperes. The addition of a trace of potassium
hydroxide to the water has been shown to facilitate the process of
dispersion. When gold wires are used, deep red colloidal solu-
tions are obtained, which after standing for several weeks, acquire
a bluish-violet color. With extra precautions, the red colloidal
gold solutions may be preserved for two years. These solutions
have been shown by Bredig to contain about 14 mg. of gold per
100 cc. In this manner Bredig prepared colloidal solutions of
platinum, palladium, iridium, and silver. The method of Bredig
has been improved and extended by Svedberg.

A diagram of Svedberg's apparatus is shown in Fig. 71. The
secondary terminals of an induction coil, capable of giving a spark
ranging from 12 to 15 cm. in length, are connected in parallel with
the electrodes and a glass plate-condenser having a surface of
approximately 225 sq. cm. Minute fragments or grains of the
metal of which a sol is desired are placed on the bottom of the
vessel containing the dispersion medium. The electrodes, which
need not necessarily be of the same metal, are immersed as shown
in the diagram, and during the process of electrical dispersion, the
contents of the vessel are gently stirred with one or the other of the

278

THEORETICAL CHEMISTRY

electrodes. With this apparatus Svedberg has succeeded in pre-
paring colloidal solutions of tin, gold, silver, copper, lead, zinc,
cadmium, carbon, silicon, selenium, and tellurium. He has also

Induction Coil

Fig. 71.

obtained all of the alkali metals in the colloidal state, ethyl ether
being used as the dispersion medium. w An interesting observation
made by Svedberg in the course of his experiments is that the color
of a metal is the same in both the colloidal and gaseous states.

CHAPTER XIII.
MOLECULAR REALITY.

The Brownian Movement. If a liquid in which fine particles
of matter are suspended, such as an aqueous suspension of gam-
boge, be examined under the microscope, the suspended particles
will be seen to be in a state of ceaseless, erratic motion. This
phenomenon was first observed in 1827 by the English botanist,
Robert Brown, while examining a suspension of pollen grains, and
has been called the Brownian Movement in honor of its discoverer.

Ever since its discovery, the Brownian Movement has been the
subject of numerous investigations. It was not until 1863, how-
ever, that Wierner suggested that the cause of the phenomenon was
the actual bombardment of the suspended particles by the mole-
cules of the suspending medium. Twenty-five years later a similar
conclusion was reached independently by Gouy, who showed that
neither light nor convection currents within the liquid could
possibly give rise to the motion. Furthermore, Gouy showed the
movement to be independent of external vibration and only slightly
influenced by the nature of the suspended particles. The smaller
the particles and the less viscous the suspending medium, the more
rapid the motion was found to be. By far the most striking feature
of the phenomenon, however, is the fact that the motion is cease-

Perrin's Experiments. The first quantitative investigation
of the Brownian Movement was undertaken by Perrin in 1909.
It has been shown (p. 100) that the mean kinetic energy Ek of one
mol of a perfect gas is given by the expression

#*=fpt>. (1)

Since pv = RT, we may write

279

280 THEORETICAL CHEMISTRY

where N denotes the Avogadro Constant, i.e., the number of
molecules contained in one mol of any gas. It is evident that if
Ek can be measured, equation (2) affords a means of calculating N,
provided we are warranted in applying an equation which has been
derived for the gaseous state to a suspension of fine particles in a
liquid medium. It has already been shown that the simple gas
laws hold for dilute solutions and therefore we may assume that,
at the same temperature, the mean kinetic energy of the dissolved
molecules is equal to that of the gaseous molecules. In other
words, at the same temperature, the mean kinetic energy of all
the molecules of all fluids is the same, and is directly proportional
to the absolute temperature. Since the gas laws apply equally
well to dilute solutions containing either large or small molecules,
Perrin held that there was no a priori reason for assuming that the
grains of a suspension should not conform to the same laws. If
this assumption be correct, the grains of a uniform suspension
should so distribute themselves under the influence of gravity
that, when equilibrium is attained, the lower layers will have a
higher concentration than the upper layers. In other words, the
distribution should be strictly analogous to the distribution of the
air over the surface of the earth, the density being greatest at the
surface and diminishing as the altitude increases.

Let us imagine a suspension to be confined within a tall
vertical cylinder whose cross-sectional area is s sq. cm. As-
suming that the suspension has come to equilibrium under the
influence of gravitation, let n be the number of grains per unit of
volume at a height h from the base of the cylinder. Since the
concentration diminishes as the height increases, the number of
grains at a height h + dh will be n — dn. The osmotic pressure
of the grains at the height h will be f nEk, where Ek is the mean
kinetic energy of each grain. In like manner, the osmotic pressure
at the height d + dh will be f (n — dn) Ek. The difference in
osmotic pressure between the two levels is — f dnEk and since the
pressure acts over a surface of s sq. cm., the difference of osmotic
forces acting over the cross-sectional area of the cylinder is
— t « dnEk. Since the system is in equilibrium, this difference in
osmotic forces must be balanced by the difference in the attraction

MOLECULAR, REALITY 281

of gravitation at the two levels. Let <£ be the volume of a single
grain, D its density, and 5 the density of the suspending medium.
The resultant downward pull upon a single grain will be <j> (D — 6)0,
where g is the acceleration due to gravity. The volume of liquid
between the two levels being s dh, it follows that the total down-
ward pull upon all the grains included between the two levels must
be nsh<t> (D — 6) g. It is this force which opposes the tendency
of the grains to distribute themselves uniformly throughout the
entire volume of the suspending medium, or, in other words, it is
the force which acts in opposition to the osmotic force — f s dnEk.
When equilibrium is established, these two forces must be equal,
and we may then write

- f s dnEk = ns dh<t> (D - 6) g. (3)

If HQ and n denote the number of grains per unit of volume at each
of two planes h units apart, we obtain, on integrating equation (3),

| Ek loge n0/n = <t> (D - 6) gh. (4)

On substituting in equation (4) the value of Ek in equation (2),
and transforming to Briggsian logarithms, we have

2.303 RT/N log n,/n = J m*g (D - 6) h, (5)

<£ being expressed in terms of the mean radius, r, of a single
grain. It is evident that if we can measure n, no, D, and r in
equation (5), the calculation of the Avogadro Constant, N, be-
comes possible.

The determination of the density of the grains, D, was carried
out in two different ways with suspensions of gamboge and mastic
which had been rendered uniform by a process of centrifuging.
In the first method, the grains were dried to constant weight
at 110°, and then by heating to a higher temperature, a viscous
liquid was obtained which, on cooling, formed a glassy solid. The
density of this solid was determined by suspending it in a solution
of potassium bromide of known density.

In the second method for the determination of D, Perrin measured
the masses mi and m^ of equal volumes of water and suspension
respectively. On evaporating the suspension to dryness, the mass
Ws of suspended solid contained in m^ grams of suspension was

282 THEORETICAL CHEMISTRY

obtained. If the density of water is d, the volume of the sus-
pended grains will be

_, m\_ wfa — wiz

V =~d d '

and consequently the density of the grains will be ms/V. The
values of D obtained by these two methods were found to be in
excellent agreement.

A microscope furnished with suitable micrometers was employed
in the determination of n and n0. With the high magnification
employed, the depth of the field of view was limited: in fact, the
measurements were carried out with a microscopic slide similar
to those used for counting the corpuscles in the blood. By focus-
sing the microscope at different depths, the average number of
grains in the field of view at each level could be counted. Perrin
was able to photograph the larger grains at different levels, whereas
with the smaller grains it was necessary to reduce the field so that
relatively few grains were visible. The average number of grains
counted at any two different levels would of course give the de-
sired ratio riQ/n.

The only other quantity in equation (5) to be measured was the
average radius of the grains r. To determine this quantity,
Perrin made use of a method similar to that used by Thomson for
counting the number of electrically charged particles in an ionized
gas. Stokes has shown that the force required to impart a uniform
velocity v, to a particle of radius r, moving through a liquid
medium whose viscosity is T?, is given by the formula, 6 irqrv. If
the motion be due to gravity, as in the case of suspensions of fine
particles, obviously the foregoing expression must be equal to the
right-hand side of equation (5), or

Giyrv = $irr*(D-$)g.

From this equation the value of r can be calculated. The rate
at which the grains settled under the influence of gravity was
determined by placing a portion of the uniform suspension in a
capillary tube and observing the rate at which the suspension
cleared, care being taken to keep the temperature constant. This
method of determining r was open to the objection that Stokes'

MOLECULAR REALITY 283

law might not apply to particles as small as those of colloidal
suspensions.

In order to test the validity of Stokes' law under these conditions,
the following modification of the method for the determination of
r was introduced. It had been observed that when a suspension is
rendered slightly acid, the grains, on coming in contact with the
walls of the containing vessel, adhered, while the motion of the
grains throughout the bulk of the liquid remained unaltered. In
this way it was possible to gradually remove all of the grains from
the suspension and count them and, knowing the total volume of
suspension taken, the average number of grains per cubic centi-
meter could be calculated. If the total mass of suspended matter
is known it is an easy matter to calculate the volume of each grain,
and from this to compute the radius, r. The value of r determined
in this way was found to agree with that calculated by the first
method, thus proving the validity of Stokes' law when applied to
colloidal suspensions.

Five series of experiments carried out by Perrin with gamboge
suspensions in which several thousand individual grains were
counted, gave as a mean value of N in equation (5), 69 X 1022.
Similar experiments with mastic suspensions gave N = 70.0 X 1022.
These values, it will be seen, are in close agreement with the value
of Avogadro's Constant given on page 41. *

The Law of Molecular Displacement. The actual movements
of the individual grains of a suspension when observed under the
microscope are seen to be exceedingly complex and erratic. The
horizontal projections of the paths of three different grains in a
suspension of mastic are shown in Fig. 72, the dots representing
the successive positions occupied by the particles after intervals of
30 seconds. The straight line joining the initial and final positions
of a particle is called the horizontal displacement A, of the particle.

If the time taken by the particle to move from its initial to its
final position be t, Einstein * has shown that the mean value of the
square of the horizontal displacement of a spherical particle of
radius r ought to be

"-TO" <6)

* Zeit. Elektrochem., 14, 235 (1908).

284

THEORETICAL CHEMISTRY

72.

MOLECULAR REALITY

285

where 77 is the viscosity of the suspending medium and where the
other symbols have their usual significance.

This equation was tested by Perrin, using suspensions of gam-
boge and mastic. Some of the results obtained are given in the
following table:

VALUES OF N CALCULATED BY EINSTEIN'S EQUATION.

Suspension.

r in
microns.*

mXlO16.

No of dis-
placements

A7 XI 0-22.

Gamboge in water .

0 367

246

1500

Gamboge in 10% solution of
glycerine

0 385

290

100

Mastic in water ....

0 52

650

1000

Mastic in 27% solution of urea.

5.50

750,000

100

* The micron is one-millionth of a meter or one ten-thousandth of a centimeter.

It will be seen that notwithstanding the large variations in the
granular masses of the different suspensions recorded in the table,
the values of JV, calculated by means of Einstein's equation, are
quite concordant. Perrin gives as the mean value of all of his
experiments, N = 68.5 X 1022.

Recent Investigations of the Brownian Movement. Nord-
lund * has recently repeated Perrin's experiments, employing a
colloidal solution of mercury and an arrangement of apparatus
whereby the movements of the particles could be recorded photo-
graphically. The mean value of N derived from twelve carefully
executed experiments was 59 X 1022, the average deviation of the
results of the individual experiments from the mean being approxi-
mately 10 per cent.

The Brownian Movement in gases has been studied by Millikan f
and by Fletcher J employing a minute drop of oil as the suspended
particle. In the gaseous state, where the intermolecular distances
are greater than in the liquid state, not only are the collisions less
frequent but the mean free paths are appreciably longer. These
conditions are favorable to the study of the Brownian Movement
and offer an opportunity for the determination of the Avogadro
Constant with a high degree of accuracy. As the mean of nearly
six thousand measurements, Fletcher gives N = 60.3 X 1022, this
value being accurate to within 1.2 per cent.

* Zeit. phys. Chem., 87, 60 (1914). f Phys. Rev., i, 220 (1913).
J Ibid., 4, 453 (1914).

CHAPTER XIV
THERMOCHEMISTRY.

General Introduction. A chemical reaction is almost invari-
ably accompanied by a thermal change. In the majority of
cases heat is evolved; a violent reaction developing a large amount
of heat, while a feeble reaction develops a comparatively small
amount. Such reactions are said to be exothermic. A relatively
small number of chemical reactions are known which take place
with an absorption of heat. These are termed endothermic reac-
tions. Instances of chemical reactions unaccompanied by any
thermal change are very rare and are almost wholly confined to
the reciprocal transformations of optical isomers. These facts,
which were first observed by Boyle and Lavoisier, led to the view
that the amount of heat evolved in a chemical reaction might be
taken as a measure of the chemical affinity of the reacting sub-
stances. However, with the advance of our theoretical knowledge,
it is now known that this is not true, although a parallelism
between heat evolution and chemical affinity frequently exists.

Thermochemistry is concerned with the thermal changes which
accompany chemical reactions.

Thermal Units. Heat is a form of energy, and like other
forms of energy it may be resolved into two factors; an intensity
factor, the temperature, and a capacity factor, which may be
measured in any one of several units. Among these units those
defined below are the most frequently employed.

The small calorie (cal.) is the quantity of heat required to raise
the temperature of 1 gram of water from 15° C, to 16° C. The
temperature interval is specified because the specific heat of water
varies with the temperature. The large or kilogram calorie (Cal.)
is the quantity of heat required to raise the temperature of 1000
grams of water from 15° C. to 16° C. The Ostwald or average

286

THERMOCHEMISTRY 287

calorie (K), is the quantity of heat required to raise the temper-
ature of 1 gram of water from the melting-point of ice to the
boiling-point of water under a pressure of 760 mm. of mercury.
It is approximately equal to 100 cal. or to 0.1 Cal. The joule (j),
a unit based on the C.G.S. system, is equal to 107 ergs. This
being inconveniently small is generally multiplied by 1000, giving
the kilojoule (J), which is therefore equal to 1010 ergs. The last
two units are open to the objection that their values are depend-
ent upon the mechanical equivalent of heat, any change in the
accepted value of which would involve a correction of the unit of
heat. The different capacity factors of heat energy are related
as follows: —
1 cal. = 0.001 Cal. = 0.01 K (approx.) = 4.183 j « 0.004183 J.

Thermochemical Equations. In order to represent the changes
in energy which accompany chemical reactions, an additional
meaning has been assigned to the chemical symbols. As ordina-
rily used, these symbols represent only the molecular or formula
weights of the reacting substances. In a thermochemical or
energy equation the symbols represent not only the weight in
grams expressed by the formula weights of the substances, but
also the amount of heat energy contained in the formula weight
in one state as compared with the energy contained in a standard
state. For example, the energy equation,

C + 2 0 = CO2 + 94,300 cal.,

indicates that the energy contained in 12 grams of carbon and
32 grams of oxgyen exceeds the energy contained in 44 grams of
carbon dioxide, at the same temperature, by 94,300 calories. In
writing energy equations it is very essential that we have some
means of distinguishing between the different states of aggrega-
tion of the reacting substances, since the energy content of a
substance is not the same in the gaseous, liquid, and solid states.
In the system proposed by Ostwald, ordinary type is used for
liquids, heavy type for solids, and italics for gases. Another and
more convenient system has been proposed, in which solids are
designated by enclosing the symbol or formula within square
brackets; liquids by the simple, unbracketed symbol or formula;

288

THEORETICAL CHEMISTRY

and gases by enclosing the symbol or formula within parentheses.
The above equation should, therefore, be written in the following

manner: —

|C] + (2 O) = (C02) + 94,300 cal.

Thermochemical Measurements. In order to measure the

number of calories evolved or ab-
sorbed when substances react, it is
necessary that the reaction should
proceed rapidly to completion. This
condition is fulfilled by two classes
of processes. In the first class we
may mention the processes of solu-
tion, hydration, and neutralization;
and in the second class, the process
of combustion.

The apparatus used for measur-
ing the capacity factor of heat energy
is a calorimeter. This instrument
may be given a variety of forms,
depending upon the particular use
to which it is to be put. A simple
form of calorimeter is shown in Fig.
73. It consists of two concentric
metal cylinders, A and 5, insulated
from each other by an air jacket,
the inner vessel being supported
en vulcanite points. Through a
vulcanite cover passes a thin walled
test tube, in which the reaction is
allowed to take place. An accurate thermometer and a ring-stirrer
also pass through the cover of the calorimeter. In order to deter-
mine the thermal capacity of the calorimeter, B is nearly filled with
water, and a known mass of water, w, at a temperature t\ is intro-
ducedintoC. Let the initial temperature of the water in B be £2- The
water in Bis stirred until the contents of both B and C have acquired
the same temperature, fe. When thermal equilibrium has been estab-
lished, it is evident that m (ti — £3) calories are required to raise

Fig. 73.

THERMOCHEMISTRY 289

the temperature of the apparatus and the water in 5, (fe — £3)
degrees. From this data it is an easy matter to calculate the
number of calories required to raise the temperature of the appar-
atus and water in B 1 degree, this being the thermal capacity of
the apparatus. The calorimeter may now be used to determine
the heat evolved or absorbed in a reaction. Suppose, for exam-
ple, that it is desired to measure the heat of neutralization of an
acid by a base. Equivalent quantities of both acid and base are
dissolved in equal volumes of water, care being taken to make the
solutions dilute. A definite volume of one solution is introduced
into C and an equal volume of the other solution is placed in a
vessel from which it can be quickly and completely transferred to
C. When both solutions have acquired the same temperature,
the thermometer in B is read and then the two solutions are
mixed. When the reaction is complete, the temperature of the
water in B is again noted. If the thermal capacity of the calori-
meter is Q, and the rise in temperature produced by the reaction
is 0, then Qd is the amount of heat evolved by the reaction. To
this quantity of heat must be added the number of calories re-
quired to raise the temperature of the products of the reaction B
degrees. The solutions of the products being dilute, their specific
heats may be assumed to be equal to unity. From the total quan-
tity of heat so obtained, the number of calories evolved when mo-
lecular quantities react can be readily calculated. The chief source
of error in calorimetric measurements is loss by radiation. This
may be reduced to a minimum (1) by making the thermal capac-
ity of the calorimeter large, and (2) by so arranging matters that
the initial temperature of the water in the calorimeter is as much
below the temperature of /the room as the final temperature is
above it.

The Combustion Calorimeter. The combustion of many sub-
stances, such as organic compounds, proceeds very slowly in air
under ordinary pressures. Such reactions can be accelerated, if
they are caused to take place in an atmosphere of compressed
oxygen. For this purpose the combustion calorimeter was de-
vised by Berthelot.* In this apparatus the essential feature is
* Ann. Chim. Phys., (5), 23, 160 (1881); (6), 10, 433 (1887).

290

THEORETICAL CHEMISTRY

the so-called combustion bomb, shown in Fig. 74. This consists
of a strong steel cylinder lined with platinum or gold, and fur-
nished with a heavy threaded cover. The substance to be
burned is placed in a platinum capsule fastened to the support
R, and a short piece of fine iron wire of known mass is connected
with the electric terminals Z, Z, the middle portion of the wire
dipping into the substance. The cover is then screwed down
tight, and the bomb is filled with oxygen under a pressure of from
20 to 25 atmospheres. The bomb is then
submerged in the calorimeter, as shown in
Fig. 75. The mass of water in the calorim-
eter being known and its temperature having
been read, an electric current is passed through
|S the iron wire in the bomb causing it to burn
and thus ignite the substance. The rise in
temperature due to the combustion is ob-
served, and the quantity of heat evolved is
calculated. Corrections must be applied for
loss by radiation, for the heat evolved from
the combustion of the iron, and for the heat
evolved from the oxidation of the nitrogen of
the residual air in the bomb.

For the details of the method of determin-
ing heats of combustion the student must
consult a laboratory manual.
Law of Lavoisier and Laplace. In 1780, Lavoisier and Laplace,*
as a result of their thermochemical investigations, enunciated the
following law: — The quantity of heat which is required to decompose
a chemical compound is precisely equal to that which was evolved in
the formation of the compound from its elements. This first law of
thermochemistry will be seen to be direct corollary of the law
of the conservation of energy which was first clearly stated by
Mayer in 1842.

Law of Constant Heat Summation. A generalization of funda-
mental importance to the science of thermochemistry was discov-
ered in 1840 by Hess.f He pointed out that the heat evolved in a

* Oeuvres de Lavoisier, Vol. II, p, 283.
t Pnffff Ann.. «n. 3ftfi HfttfN.

Fig. 74.

THERMOCHEMISTRY

291

chemical process is the same whether it takes place in one or in several
steps. This is known as the law of constant heat summation. The

Fig. 75.

truth of the law may be illustrated by the equality of the heat of
formation of ammonium chloride in aqueous solution, when pre-
pared in two different ways.

292 THEORETICAL CHEMISTRY

Thus,

(A)

(NH3) + (HC1) = [NH4C1] + 42,100 cal,

[NH4C1] + aq. = NH4C1, aq. - 3,900 cal.

38,200 cal.

(B)

(NH3) + aq. = NH3, aq. + 8,400 cal.

(HC1) + aq. = HC1, aq. + 17,300 cal.

NH3, aq. + HC1, aq. = NH4C1, aq. + 12,300 cal.

38,000 cal.

It will be observed that the total amount of heat evolved in
the formation and solution of ammonium chloride is the same
within the limits of experimental error, whether gaseous ammonia
and hydrochloric acid are allowed to react and the resulting prod-
uct is dissolved in water, or whether the gases are each dissolved
separately and then allowed to react. It should be noted that
when a substance is dissolved in so much water that the addition
of more water or the removal of a small portion of water produces
no thermal effect, it is customary to denote it by the symbol aq.
(Latin aqua = water). Thus,

NH4C1, aq. + nH20 = NH4C1, aq.,
NH4C1, aq. - nH2O = NH4C1, aq.

By means of the law of constant heat summation it is possible to
find indirectly the amount of heat developed or absorbed by any
reaction, even though it is impossible to carry it out experimen-
tally. For example, it is impossible to measure the heat evolved
when carbon burns to carbon monoxide. But the heat evolved
when carbon monoxide burns to carbon dioxide, and also the heat
evolved when carbon burns to carbon dioxide, can be accurately
determined. The energy equations are as follows: —

[C] + 2(0) = (CO,) + 94,300 cal. (1)

(CO) + (0) = (C02) + 67,700 cal. (2)

THERMOCHEMISTRY 293

'Treating these equations algebraically, and subtracting equation
(2) from equation (1), we have

[C] + (0) « (CO) + 26,600 cal.,

or, the heat of combustion of carbon to carbon monoxide is 26,600
calories. Again, as a further illustration of the applicability of
the law of Hess, we may take the calculation of the heat of forma-
tion of hydriodic acid from its elements, making use of the follow-
ing energy equations: —

2 KI, aq. + 2 (Cl) = 2 KC1, aq. + 2 [I] + 524 K (1)

2 HI, aq. + 2 KOH, aq. = 2 KI, aq. + 2 H2O + 274 K (2)

2 HC1, aq. + 2 KOH, aq. = 2 KC1, aq. + 2 H20 + 274 K (3)

2 (HI) + aq. = 2 HI, aq. + 384 K, (4)

2 (HC1) + aq. = 2 HC1, aq. + 346 K, (5)

2 (H) + 2 (Cl) = 2 (HC1) + 440 K, (6)

adding equations (1) and (2),

2 (Cl) + 2 HI, aq. + 2 KOH, aq. = 2 KC1, aq. + 2 [I] + 2 H2O

+ 798 K. (7)

Subtracting equation (3) from equation (7),

2 (Cl) + 2 HI, aq. - 2 HC1, aq. = 2 [I] + 524 K,
or

2 (Cl) + 2 HI, aq. = 2 [I] + 2 HC1, aq. + 524 K, (8)

adding equations (4) and (8),

2 (HI) + aq. + 2 (Cl) - 2 [H + 2 HC1, aq. + 908 K, , (9)
subtracting equation (5) from equation (9),

2 (HI) + 2 (Cl) - 2 (HC1) = 2 [I] + 562 K,
or

2 (HI) + 2 (Cl) = 2 [I] + 2 (HC1) + 562 K, (10)

subtracting equation (6) from equation (10),

2 (HI) -2(H) = 2[I] + 122K,
or

2 (H) + 2 (I) = 2 (HI) - 122 K.

In a similar manner, practically any heat of formation may be
calculated, provided the proper energy equations are combined.

294 THEORETICAL CHEMISTRY

Heat of Formation. The intrinsic energy of the substances
entering into chemical reaction is unknown, the amount of heat
evolved or absorbed in the process being simply a measure of the
difference between the energy of the reacting substances and the
energy of the products of the reaction. Thus, in the equation

[C] + 2 (O) = (C02) + 94,300 cal.,

the difference between the energy of a mixture of 12 grams of
carbon and 32 grams of oxygen, and the energy of 44 grams of
carbon dioxide is seen to be 94,300 calories. The equation is
clearly incomplete since we have no means of determining the
intrinsic energies of free carbon and oxygen. Furthermore, since
the elements are not mutually convertible, we have no means of
determining the difference in energy between them. It is cus-
tomary, therefore, in view of this lack of knowledge, to put the
intrinsic energies of the elements equal to zero.

If the heats of formation of the substances present in a reac-
tion are known, it is much simpler to substitute these in the
energy equation and solve for the unknown term. This method
avoids the laborious process of elimination from a large number of
energy equations, as in the preceding pages. If all of the sub-
stances involved in a reaction are considered as decomposed into
their elements, it is evident that the final result of the reaction
will be the difference in the sums of the heats of formation on the
two sides of the equation. This leads to the following rule: —
To find the quantity of heat evolved or absorbed in a chemical reac-
tion, substract the sum of the heats of formation of the substances
initially present from the sum of the heats of formation of the products
of the reaction j placing the heat of formation of all elements equal to
zero.

The energy equation for the formation of carbon dioxide from
its elements may y.hen be written as follows: —

0 + 0= (C02)+ 94,300 cal.,
or

(C02) = -94,300 cal.

That is, the energy of 1 mol of carbon dioxide is —94,000 calories.
Therefore in writing an energy equation we make use of the fol-

THERMOCHEMISTRY 295

lowing rule: — Replace the formulas of each compound in the equa-
tion representing the reaction by the negative values of their respective
heats of formation and solve for the unknown term. This unknown
term may be either the heat of a reaction or the heat of formation
of one of the reacting substances. The following examples will
serve to illustrate the application of the above rules: —

(1) Let it be required to find the heat of the following reaction

[MgCl2] + 2 [Na] - 2 [NaCl] + [Mg] + x,

where x is the heat of the reaction. The heat of formation of
MgCl2 is 151 Cal., and that of NaCl is 97,9 Cal., therefore,

- 151 + 0 = - (2 X 97.9) + 0 + x,
or

x = 44.8 Cal.

(2) The heat of combustion of 1 mol of methane is 213.8 Cal.,
and the heats of formation of the products, carbon dioxide and
water, are 94.3 Cal. and 68.3 Cal., respectively. Let it be required
to find the heat of formation of methane. Representing the heat
of formation of methane by x, we have

(CH4) + 2 (02) = (C02) + 2 (H20) + 213.8 Cal.,

- x + o = - 94.3 - 2 X 69.3 + 213.8,
or

x = 17.1 Cal.

(3) The heat of combustion of 1 mol of carbon disulphide is
265.1 Cal., the thermochemical equation being

' CS2 + 3 (02) = (C02) + 2 (S02) + 265.1 Cal.
The heats of formation of carbon dioxide and sulphur dioxide are
94.3 Cal. and 71 Cal. respectively. The heat of formation of
carbon disulphide x, may then be calculated as follows: —

_ x + o = - 94.3 - 2 X 71 + 265.1,
or

a; =*f28.8 Cal.

Carbon disulphide is thus seen to be an endothermic compound.

Heat of Solution. The thermal change accompanying the

solution of 1 mol of a substance in so large a volume of solvent

that subsequent dilution of the solution causes no further thermal

296

THEORETICAL CHEMISTRY

change is termed the heat of solution. The solution of neutral
salts is generally an endothermic process. This fact may be
readily accounted for on the hypothesis that considerable heat

HEATS OF FORMATION AND SOLUTION. UT\

Substance.

Hoat of
Formation,

Heat of
Solution.

Water, vapor

58.7

Water, liquid

68 4

Hydrochloric acid

22 0

20.3

Sulphuric acid

193 1

17 8

Ammonia

12 0

8 4

Nitric acid

41 9

7 2

Phosphoric acid. .

302 9

2 7

Potassium hydroxide

103 2

13.3

Potassium chloride

104 3

-3.1

Potassium bromide

95 1

-5.1

Potassium iodide

80 1

-5 1

Potassium nitrate

119.5

-8 5

Sodium hydroxide

101 9

10 9

Sodium chloride

97.6

1.2

Sodium bromide

85 6

-0 2

Sodium sulphate

328 8

0 2

Sodium nitrate ...

111 3

-5 0

Sodium carbonate

272 6

5 6

Ammonium chloride

75 8

-4 0

Ammonium nitrate

88 0

-6 2

Calcium hydroxide

215 0

3.0

Calcium chloride

170 0

17 4

Magnesium sulphate

502 0

20.3

Ferrous chloride

82 0

17 9

Ferric chloride

96.1

63.3

Zinc chloride

97 0

15.6

Zinc sulphate

30 0

18.5

Cadmium chloride

293.2

3 0

Cupric chloride

51 6

11.1

Cupric sulphate

182 6

15 8

Mercuric chloride

53.2

-3.3

Silver nitrate

28.7

-5.4

Stannous chloride

80 8

0.3

Stannic chloride . .

127.3

'29.9

Lead chloride

82.8

-6.8

Lead nitrate

105 5

-7.6

must be absorbed as heat of fusion and heat of vaporization before
the solid salt can assume a condition in solution which closely
resembles that of a gas. The heat of solution of hydrated salts
is less than the heat of solution of the corresponding anhydrous

THERMOCHEMISTRY

297

salts. For example, the heat of solution of 1 mol of anhydrous
calcium nitrate is 4000 calories, while the heat of solution of 1 mol
of the tetrahydrate is —7600 calories. The difference between
the heats of solution of the anhydrous and hydrated salts is termed
the heat of hydration. The heats of formation and heats of solu-
tion in water of some of the more common compounds are given
in the preceding table, the values being expressed in large calories.

Heat of Dilution* The heat of dilution of a solution is the
quantity of heat per mol of solute which is evolved or absorbed
when the solution is greatly diluted. Beyond a certain dilution,
further addition of solvent produces no thermal change. While
there is a definite heat of solution for a particular solute in a par-
ticular solvent, the heat of dilution remains indefinite, since the
latter is dependent upon the degree of dilution. Those gases
which obey Henry's law are practically the only substances which
have no appreciable heats of solution or dilution.

The following tables give the heats of dilution of hydrochloric
and nitric acids.

HEAT OF DILUTION OF SOLUTIONS OF HYDROCHLORIC

ACID.

Heat of solution = 20.3 cal.

HC1+H2O 5.37

HC1+2H2O 11.36

HC1+10H20 16.16

HC1+50H20 17.1

HC1+300H20 17.3

HEAT OF DILUTION OF SOLUTIONS OF NITRIC ACID.
Heat of solution = 7.15 cal.

HNO34-H2O

3 84

HN03-f-2H2O

2 32

HNO3+4H2O

1 42

HNO8-*-6H20

0.2

HN03-f8H2O

-0 04

HNO3-t-100H2O

-0 03

298 THEORETICAL CHEMISTRY

Reactions at Constant Volume. When a chemical reaction
takes place without__any change in volum^ or when the external
work resulting from a change in volume is not included in the
heat of the reaction, the process is said to take place at constant
volume. That is to say, the condition of constant volume is a
condition which involves no^ external jvprk^ either positive or
negative. Under these conditions the total energy of the react-
ing substances is equal to the total energy of the products of the
reaction, plus the quantity of heat developed by the reaction.

Reactions at Constant Pressure. When a chemical reaction
is accompanied by a change in volume, the system suffers a loss
of heat equivalent to the work done against the atmosphere, if
the volume increases; or the system gains an amount of heat
equivalent to the work done on the system by the atmosphere,
if the volume decreases. Under these conditions the reaction is
said to take place at constant pressure. The difference between
constant volume and constant pressure conditions, then, is that
under the former, the heat equivalent of the work corresponding
to any change in volume which may occur is not considered as
having any effect upon the energy of the system; whereas under
the latter, due account is taken of the change in energy resulting
from change in volume. Suppose that in a reaction, 1 mol of gas
is formed. Under standard conditions of temperature and
pressure the volume of the system will be increased by 22.4 liters.
The formation of gas involves the performance of work against
the atmosphere, this work being done at the expense of the heat
energy of the system. To calculate the heat equivalent of the
work done, let us imagine the gas enclosed in a cylinder fitted with
a piston whose area is 1 square centimeter. The normal pressure
of the atmosphere on the piston is 76 cm. of mercury or 1033.3
grams per square centimeter. If the increase in the volume of
the gas is 22.4 liters, the piston will be raised through 22,400 cm.
and the work done will be 1033.3 X 22,400 gram-centimeters.
The heat equivalent of this change in volume will be (1033.3 X
22,400) -5- 42,600 = 542.3 calories or 0.5423 large calories. This
amount of heat must be added to the heat of the reaction. It
should be observed that this correction is independent of the actual

THERMOCHEMISTRY 299

value of the pressure upon the system. Thus, if the pressure is
increased n times, the volume of the gas will be reduced to 1/n
of its former value, and the work done will involve moving the
piston through 1/n of the distance against an n-fold pressure,
which is plainly equivalent to the former amount of work. While
the correction is independent of the pressure it is not independent
of the temperature. The familiar equation, pv ^RT± shows us
that the^work^done by a gas is directly proportional to its ab-
solut^iaaiperatureT Thus, if a gas is evolved at 273°, it will
occupy double the volume it would occupy at 0°, and the work
done at 273° will involve moving the piston through twice the
distance that it would have to be moved at 0°. Theoretically, a
gas evolved at the absolute zero would occupy no volume and
hence no work would be done. Introducing the correction for
temperature, we see that

= 1.986 Teal.,

must be added to the heat of the reaction, where T is the absolute
temperature at which the change in volume occurs. For all
ordinary purposes it is sufficiently accurate to take 2 T calories
as the correction. Thus, suppose n mols of gas to be formed in a
reaction at 17° C., the amount of heat absorbed will be

nx 2(273 + 17) = 580ncal.

Under constant pressure conditions, the symbols, in addition to
their usual significance, represent the energy plus or minus the
term, 2 T per mol, the positive or negative sign being used
according as the gas is absorbed or formed. Since the constant
volume condition is a condition in which no account is taken of the
external work, even if a change in volume does occur during the
reaction, and the constant pressure condition is one in which
the external work is taken into consideration, it is apparent that
the relation of the heat energy of a reaction at constant volume,
QV) to the heat energy at constant pressure QP) can be represented
by the equation

Qp = Qv-2n!TcaL,

300 THEORETICAL CHEMISTRY

where n denotes the number of mols of gas formed in excess of
those initially present. This equation is of great importance in
connection with the determination of heats of combustion in the
bomb-calorimeter in which the reactions necessarily take place
under constant volume conditions. Since it is customary to state
heats of reaction under constant pressure conditions, the foregoing
equation makes it possible to convert heats of combustion deter-
mined under constant volume conditions into heats of combustion
under constant pressure conditions. For example, the combustion
of naphthalene takes place in accordance with the equation

CioH8 + 12 (02) = 10 (C02) + 4 (H20) + 1242.95 Cal.

It is apparent that the combustion is accompanied by the forma-
tion of 2 mols of gas, and at 15° C. the correction will be

Qp = 1242.95 - [2 X 0.002 (273 + 15)],
or

Qp= 1241.8 Cal.

The volume occupied by solids or liquids is so small as to be
negligible and does not enter into these calculations.

Variation of Heat of Reaction with Temperature. If a chem-
ical reaction be allowed to take place first at the temperature fo,
and then at the temperature k, the amounts of heat developed in
the two cases will be found to be quite different. Let Qi and Q2
represent the quantities of heat evolved at the temperatures ti
and t% respectively. Let us imagine that the reaction takes place
at the temperature ti, Qi units of heat being evolved; and then
let the products of the reaction be heated to the temperature fe.
If c' represents the total thermal capacity of the products of the
reaction, then the quantity of heat necessary to produce this rise
in temperature will be c' (h — h). Now let us imagine the
reacting substances, at the temperature h, to be heated to the
temperature £2, and then allowed to react with the evolution of
Q2 units of heat. The heat necessary to produce this rise in
temperature in the reacting substances is c (fe — 2i), where c
is the total thermal capacity of the original substances. Having
started with the same substances at the same initial temperature,
and having obtained the same products at the same final temper-

THERMOCHEMISTRY 301

ature, we have, according to the law of the conservation of
energy,

Qi-c'(fe-fe) -&-c(fe-«i),
or

or, where the change in temperature is very small,

, ~ — /» — /»'

df " C °'

If c' is greater than c then the sign of dQ/dt will be negative, or, in
other words, an increase in temperature will cause a decrease in
the heat of reaction. On the other hand, if c is greater than c',
dQ/dt will be positive and the heat of reaction will increase with
the temperature.

EXAMPLE. The reaction between hydrogen and oxygen at
18° C. is represented by the following equation: —

2 (H2) + (02) = 2 (H2O) + 1367.1 K

Suppose it is required to find how much heat will be evolved when
equal masses of the two gases react at 110° C., the product of the
reaction being maintained at this temperature, and the pressure
remaining constant. The specific heats per gram of the different
substances involved are as follows: —

Hydrogen = 3.409; Oxygen = 0.2175; Water (between 18°
and 100°) = 1; (Water between 100° and 110°) = 0.5.
The heat of vaporization of water is 537 calories per gram.

For liquid water per degree we have,

dQ/dt = (4 X 3.409 + 32 X 0.2175) - (36 X 1) = - 15.404 cal.

and for (100° -18°) = 82°, we have, 82 x(- 15.404) = -1263 cal.
The heat of formation of liquid water at 100° is, therefore,
1367.1- 12.63 = 1354.47 K.

When the liquid water is vaporized at 100°, (36 X 537) calories of
this heat is absorbed, or the formation of steam at 100° from
hydrogen and oxygen, evolves

1354.47 - 193.32 = 1161.15 K.

302

THEORETICAL CHEMISTRY

For steam per degree, we have,

dQ/dt = (4 X 3.409 + 32 X 0.2175) - (36 X 0.5) = 2.596 cal;,
and for the interval (110° - 100°) = 10°,

10 X 2.596 = 25.96 cal.
Or for the total heat evolved, we have

1161.15 + 0.2596 = 1161.41 K.

Heat of Combustion. The heat evolved during the complete
oxidation of unit mass of a substance is termed its "heat of
combustion. The unit of mass commonly chosen in all physico-
chemical calculations is the mol. An enormous amount of
experimental work has been done by Thomsen,* Berthelot,t and
Langbein t on the determination of the heats of combustion of a
large number of organic compounds. A few of their results are
given in the accompanying tables.

SATURATED HYDROCARBONS.

Hydrocarbon.

Heat, of
Combustion.

Difference.

Methane CHU .

Cal.
211 9

Cal.

Ethane C2EU . ...

370.4

158 5

Propane, C$Hs

529 2

158 8

Butane, C4Hio .

687 2

158 0

Petane, CsH^

847 1

159.9

UNSATURATED HYDROCARBONS.

Hydrocarbon.

Heat of
Combustion

Difference.

Ethylene, C2H4

Cal.

333 4

Cal.

1 CA O

Propylene, CsHU

492 7

159 3

Isobutylene, C^s

650 6

157.9

Amylene, CsHio

807 6

157 0

Acetylene, C2HU

310 1

1 C*7 C

Allylene, C3H4. . ...

467.6

157 5

* Thermochemische Untersuchungen, 4 Vols.

t Essai de Mecanique Chimique, Thermochimie, Done*es et Lois Numeri-
ques.

f Jour, prakt, Chem., 1885 to 1895.

THERMOCHEMISTRY

303

ALCOHOLS.

Alcohol.

Heat of
Combustion.

Difference.

Methyl alcohol, CH*O

Cal.
182 2

Cal.

Ethyl alcohol, C2H6O

340 5

158.3

Propyl alcohol, CsHsO

498 6

158.1

Isobutyi alcohol, CiHioO

658 5

159.9

It will be observed that a very nearly constant difference in
the heat of combustion corresponds to a constant difference of a
CH2 group in composition. A number of interesting relations
between heats of combustion of compounds and their differences
in composition have been discovered, but these cannot be taken up
at this time. It has also been pointed out that the heat of com-
bustion of organic compounds is conditioned not only by their
composition, but also by their molecular constitution.

Some exceedingly interesting and important results have been
obtained with the different allotropic forms of the elements. For
example, when equal masses of the three common allotropic forms
of carbon are burned in oxygen, the amounts of heat evolved are
found to be quite different, as is shown by the following energy
equations: —

[C] diamond + 2 (0) - (C02) + 94.3 Cal.
[C] graphite + 2 (0) = (CO*) + 94.8 Cal.
[C] amorphous + 2 (0) = (C02) + 97.65 Cal.

It is apparent that amorphous carbon contains the greatest
amount of energy of any one of the three allotropic modifications,
and, therefore, when amorphous carbon is changed into diamond,
the reaction must be accompanied by the evolution of (97.65 —
94.3) = 3.35 Cal. In like manner, the allotropic forms of sulphur
and phosphorus have different heats of combustion. The follow-
ing equations show the heat equivalents of the differences in
intrinsic energy between the allotropic forms: —

S (monoclinic) = S (rhombic) + 2.3 Cal.
P (white) = P (red) + 3.71 Cal.

304 THEORETICAL CHEMISTRY

When the same substance is burned in oxygen and then in ozone,
it is found that more heat is evolved in ozone than in oxygen.
The energy equation expressing the change of ozone into oxygen
may be written thus,

(03) = lJ(Q») + 36.2Cal.

All of the above facts illustrate the general principle that the
larger amounts of intrinsic energy are associated with the more
unstable forms.

Thermoneutrality of Salt Solutions. In addition to the law
of constant heat summation, Hess discovered two other important
laws of thermochemistry, viz., the law of thermoneutrality of
salt solutions, and the law governing the neutralization of acids
by bases.* When two dilute salt solutions are mixed there is
neither evolution nor absorption of heat. Thus when dilute solu-
tions of sodium nitrate and potassium chloride are mixed, there is
no thermal effect. The energy equation may be written as follows : —

NaN03, aq. + KC1, aq. = NaCl, aq. + KNOd, aq. + 0 Cal.

According to this equation a double decomposition has taken
place and we should naturally expect an evolution or an absorption
of heat. While Hess could not account for the absence of any
thermal effect, he recognized the fact as quite general and formu-
lated the law of the thermoneutrality of salt solutions as fol-
lows: — The metathesis of neutral salts in dilute solutions takes place
with neither evolution nor absorption of heat.

The explanation of the phenomenon of thermoneutrality was
furnished by the theory of electrolytic dissociation. When the
above equation is written in the ionic form, it becomes

Na* + N(Y + 1C + Cl' - Na- + Cl' + K' + N(Y.

From this it is apparent that the same ions exist on both sides of
the equation, and in reality no reaction takes place.

There are numerous exceptions to the law of thermoneu-
trality/ These can be satisfactorily accounted for by the theory of
electrolytic dissociation. All of those salts, the behavior of which in
dilute solution is contrary to the law, are found to be only partially

* Pogg. Ann., 50, 385 (1840).

THERMOCHEMISTRY 305

ionized, and, therefore, when their solutions are mixed, a chem-
ical reaction actually occurs. The exceptions must be considered
as furnishing additional evidence in favor of the theory of elec-
trolytic dissociation.

Heat of Neutralization. Hess also discovered * that when
dilute solutions of equivalent quantities of strong acids and
strong bases are mixed, practically the same amount of heat is
evolved. The following energy equations may be considered as
typical examples of such neutralizations: —

HC1, aq. + NaOH, aq. = NaCl, aq. + H20 + 13.75 CaL,
HN03, aq. + NaOH, aq. = NaN03 aq. + H20 + 13.68 CaL,
HC1, aq. + KOH, aq. = KC1, aq. + H20 + 13.70 CaL,
HN03, aq. + KOH, aq. = KNO3, aq. + H20 + 13.77 CaL,
HC1, aq. + LiOH, aq. = LiCl, aq. + H20 + 13.70 CaL

Here again it would be difficult to explain the phenomenon with-
out the theory of electrolytic dissociation. In terms of this
theory, however, the explanation is perfectly plausible. If MOH
and HA represent any strong base and any strong acid respectively,
then when equivalent amounts of these are dissolved in water,
each solution being largely diluted to the same volume, the reac-.
tion may be written thus: —

M' + OH' + H* + A' = M* + A' + H20 + 13.7 CaL

Disregarding the ions which occur on both sides of the equality
sign, we have

H' = H20 + 13.7Cal.

It thus appears that the neutralization of a strong acid by a strong
base in dilute solution consists solely in the combination of hydro-
gen and hydroxyl ions to form undissociated water, the heat of
this ionic reaction being 13.7 large calories.

The heat of formation of water from its ions must not be con-
fused with the heat of formation of water from its elements.
When weak acids or weak bases are neutralized by strong bases
or strong acids, or when weak acids are neutralized by weak bases,

* Loc. cit.

306 THEORETICAL CHEMISTRY

the heat of neutralization may differ widely from 13.7 Cal. This
is shown by the following thermochemical equations: —

H-COOH, aq. + NaOH, aq. = H-COONa, aq. + H20

+ 13.40 Cal.,
CHC12-COOH, aq. + NaOH, aq. = CHCl2-COONa, aq. + H2O

+ 14.83 Cal.,
H-COOH, aq. + NH4OH, aq. = H-COONH4, aq. + H20

+ 11.90 Cal.,
HCN, aq. + NaOH, aq. = NaCN, aq. + H20 + 2.90 Cal.

As will be seen, the heat of neutralization may be either greater
or less than 13.7 Cal. The exceptions to the generalization of
constant heat of neutralization are readily explained by the
theory of electrolytic dissociation. Suppose a weak acid to be
neutralized by a strong base. According to the dissociation
theory, the acid is only slightly dissociated and, therefore, yields
a comparatively small number of hydrogen ions to the solution.
The base on the other hand is completely dissociated into hydroxyl
and metallic ions. Therefore, as many hydroxyl ions disappear
as there are free hydrogen ions with which they can combine to
form water. When the equilibrium between the acid and the
products of its dissociation has been thus disturbed, it undergoes
further dissociation and the resulting hydrogen ions immediately
combine with the free hydroxyl ions of the base. This process
continues until all of the hydroxyl ions of the base have been
neutralized. It is evident that the thermal effect in this case is
the algebraic sum of the heat of dissociation of the weak acid,
which may be positive or negative, and the heat of formation of
water from its ions. A similar explanation holds for the neutrali-
zation of a weak base by a strong acid, or for the neutralization
of a weak acid by a weak base. This affords a method for estimat-
ing the approximate value of the heat of dissociation of a weak
acid or a weak base. For example, in the equation given above,

HCN, aq. + NaOH, aq. = NaCN, aq. + H20 + 2.90 Cal.,

the difference between 2.9 and 13.7 or -10.8 Cal. represents
approximately the heat of dissociation of hydrocyanic acid.

THERMOCHEMISTRY

307

Since the acid is initially slightly dissociated in dilute solution,
it is apparent that in order to obtain the true heat of dissociation
we must add to —10.8 Cal. the thermal value of the dissociation
of that portion of the acid which has already become ionized.
Heat of lonization. Since 13.7 Cal. is the heat of formation
of water from its ions, this must also be the thermal equivalent of
the energy required to dissociate one mol of water into its ions. It
must be remembered that the dissociated molecule of water must
be mixed with a very large volume of undissociated water, in order

HEAT OF FORMATION OF IONS.

Ion.

Heat of
Formation.

Ion.

Heat of
Formation

Hydrogen

0 0

j Copper (ic)

-15 8

Potassium
Sodium ...
Lithium . ...
Ammonium
Magnesium
Calcium ....

61 9
57 5
62 9
32 8
109 0
109 0

i Copper (ous) . .
Mercury (ous)
Silver ... ....
Lead .
Tin (ous).
Chlorine . . . .

-16 0

-19 8
-25 3
0 5
3 3
39 3

Aluminium

121 0

Bromine

28 2

Manganese . .
Iron (ous)
Iron (ic)
Cobalt
Nickel
Zinc

50 2
22 2
-9 3
17 0
16 0
35 1

Iodine
Sulphate
Sulphite
Nitrous
Nitric
Carbonate

13 1
214 4
151 3
27 0
49 0
161 1

Cadmium

18 4

Hydroxyl . ....

54 7

that the dissociation may be permanent. Reference to the table
of heats of formation (p. 296), will show that 68.4 Cal. are required
to form one mol of water from its elements. Hence, it follows that
68.4 — 13.7 = 54.7 Cal., is the heat of formation of one equivalent
of hydrogen and hydroxyl ions. It has been shown that an ex-
tremely small amount of energy is necessary to ionize hydrogen
when it is dissolved in water. It is evident, therefore, that 54.7
Cal. is a close approximation to the heat of formation of one equiva-
lent of hydroxyl ions.

On the assumption that the heat of ionization of gaseous hydro-
gen in solution is zero, the values of the other ionic heats of forma*

308 THEORETICAL CHEMISTRY

tion may be computed. For example, the heat of formation of
KOH, aq. is 116.5 Cal. The ionic heat of formation of potassium
ions must be 116.5 - 54.7 = 61.8 Cal. In like manner, the
heat of formation of KC1, aq. is 101.2 Cal.; hence the ionic heat
of formation of chlorine ions must be 101.2 — 61.8 = 39.4 Cal.
The preceding table of ionic heats of formation has been calculated
as in the above examples.

The Principle of Maximum Work. A fundamental principle
of the science of mechanics is, that a system is in stable equilib-
rium when its potential energy is a minimum. In 1879, Ber-
thelot * suggested that a similar principle applies to chemical
systems.

In terms of the kinetic theory, the temperature of a substance
is to be regarded as a measure of the kinetic energy of its molecules.
The development of heat by a chemical reaction would, therefore,
be taken as an indication of a decrease in the potential energy of
the system. Berthelot's theorem, known as the principle of
maximum work, may be stated as follows: — " Every chemical proc-
ess accomplished without the intervention of any external energy
tends to produce that substance or system of substances which evolves
the maximum amount of heat.11 The table of heats of formation
(p. 296) illustrates the general truth of this principle, but as will
be seen, the theorem precludes the possibility of spontaneous
endotherrnic reactions. Thus, for example, the formation of
acetylene* from its elements at the temperature of the electric
arc is a well-known endothermic reaction, but according to the
principle of maximum work, it would not take place spontaneously.
Another serious objection to Berthelot's principle is, that accord-
ing to it, all chemical reactions should proceed to completion, the
reaction taking place in such a way as to evolve the greatest amount
of heat. As is well known, many reactions, and theoretically all
reactions, are never complete, but proceed until a condition of
equilibrium is reached. The principle of maximum work, there-
fore, denies the existence of equilibria in chemical reactions.
Many attempts have been made to " explain away " these defects,
but none of them have been successful. In referring to the generali-

* Essai de Mecanique Chirmque.

THERMOCHEMISTRY 309

zation, Le Chatelier terms it "a very interesting approximation
toward a strictly valid generalization. "

^The Theorem of Le Chatelier. As a result of his attempts to
modify the principle of maximum work and render it generally
applicable, Le Chatelier was led to the discovery of a rigorous law
of wide-reaching usefulness. His generalization may be stated
as follows: — Any alteration in the factors which determine an equi-
librium, causes the equilibrium to become displaced in such a way
as to oppose, as far as possible, the effect of the alteration. If the
temperature of a system which is in equilibrium be raised or
lowered, the resulting displacement of the equilibrium is accom-
panied by such absorption or evolution of heat as will tend to
maintain the temperature constant. An interesting illustration
of the behavior of a system when one of the factors controlling
the equilibrium is varied, is afforded by the system

2N02<=±N204.

The reaction proceeds in the direction indicated by the upper
arrow with the evolution of 12.6 Cal. Increase of temperature
favors the reaction which is accompanied by an absorption of
heat, which in this case, is the reaction indicated by the lower
arrow. Hence as the temperature rises, the percentage of N02
increases at the expense of N2O4. This fact can be demonstrated
by the following experiment: Some liquefied N204 is placed in
each of three long glass tubes, which are sealed at one end. When
enough N204 has vaporized to displace the air, the open ends of
the tubes are sealed. Changes in the equilibrium caused by
varying the temperature can be followed by noting the changes
in the color of the mixture. N204 is an almost colorless substance,
while N02 is reddish brown. At ordinary temperatures the
contents of the tubes will be brown in color. One tube is set
aside as a standard of comparison, while the temperature of the
second is lowered by surrounding it with a freezing mixture. As
the temperature falls, the brown color of the contents of the tube
becomes much lighter, showing an increased formation of N204.
The third tube is heated by immersing it in a beaker of boiling
water. As the temperature rises, the contents of the tube becomes

SlO THEORETICAL CHEMISTRY

much darker in color, indicating an increase in the amount of N(>2
in the mixture.

Another example is afforded by the equilibrium between ozone
and oxygen, represented by the equation

The reaction indicated by the upper arrow is exothermic. In-
crease of temperature causes a displacement of the equilibrium
in the direction of the lower arrow, since under these conditions
heat is absorbed. Thus, as the temperature rises ozone becomes
increasingly stable. Nernst has calculated that at 6000° C.,
the temperature of the photosphere of the sun, 10 per cent of the
above equilibrium mixture would be ozone. Other applications
of the theorem of Le Chatelier will be given in subsequent
chapters.

PROBLEMS.

1. From the following data calculate the heat of formation of HNO2
aq.—

[NH4N02] = (N2) + 2 H20 + 71.77 Gal,

2 (H2) + (O2) = 2 H20 + 136.72 Cal,

(N2) + 3 (H2) + aq. == 2 NH, aq. + 40.64 Cal,

NH3 aq. + HN02 aq. = NH4N02 aq. + 9.110 Cal.,

[NH4N02] + aq. = NH4N02 aq. - 4.75 Cal.

Ans. (H) + (N) + (02) + aq. = HN02 aq. + 30.77 Cal

2. By the combustion at constant pressure of 2 grams of hydrogen
with oxygen to form liquid water at 17° C., 68.36 Cal. are evolved. What
is the heat evolution at constant volume? Ans. 67.49 Cal.

3. The heats of solution of NajSO*, Na2S04.H20, and Na2S04.10 H2O
are 0.46, -1.9 and -18.76 Cal. respectively. What are the heats of
hydration of Na2S04; (a) to monohydrate, (b) to decahydrate?

Ans. (a) 2.36 Cal., (b) 19.22 Cal.

4. The heats of neutralization of NaOH and NH4OH by HC1 are 13.68
and 12.27 Cal. respectively. What is the heat of ionization of NH4OH,
if it is assumed to be practically undissociated? Ans. - 1.41 CaL

THERMOCHEMISTRY 311

5. From the following energy equations: —

[C] + (02) = (CO2) + 96.96 Cal.,

2 (H2) + (O2) - 2 H2O + 136.72 Cal.,

2 CeHe + 15 (02) = 12 (C02) + 6 H20 + 1598.7 Cal.,

2 (C2H2) + 5 (O2) - 4 (CO;) + 2 H2O + 620.1 Cal.,

all at 17° C. and constant pressure, calculate the heat evolved at 17° 0.
in the reaction

(a) at constant pressure, and (b) at constant volume.

Ans. (a) 130.8 Cal., (b) 129.07 Cal.

6. Calculate the heat of formation of sulphur trioxide from the follow-
ing energy equations: —

[PbO] + [S] + 3 (0) = [PbSOJ + 1655 K.
[PbO] + H2S04.5 H2O = [PbSOJ + 6 H2O + 233 K.
[S] + 3 (O) + 6 H20 « H2S04.5 H20 + 1422 K.
[S03] + 6 H20 = H2S04.5 H20 + 411 K.

Ans. [S] + 3 (0) = [S08] + 1011 K

7. What is the heat of formation of a very dilute solution of calcium
chloride? (See table on p. 307.) Ans. 187.6 Cal.

CHAPTER XV.
HOMOGENEOUS EQUILIBRIUM.

Historical Introduction. In this and the two succeeding
chapters, the conditions which affect the rate and the extent of
chemical reactions will be considered. When two substances
react chemically, it is customary to refer the phenomenon to the
existence of an attractive force known as chemical affinity.

Ever since the metaphysical speculations of the Greeks, who
endowed the atoms with the instincts of love and hate, the nature
of chemical affinity has been under discussion. So little has been
learned as to the pause of chemical reactions, that in recent years
this question has been dismissed and attention has been directed
to the more promising question as to how they take place. New-
ton's discovery of the law of gravitation led him to consider the
attraction between atoms and the attraction between large masses
of matter as manifestations of the same force.

Although Newton found that chemical attraction does not
follow the law of the inverse square, yet his suggestion exerted
a profound influence upon the minds of his contemporaries.

Geoffroy and Bergmann arranged chemical substances in the
order of their displacing power. Thus, if we have three sub-
stances, Ay B, and C and the attraction between A and B is
greater than that between A and C, then when B is added to AC
it will completely displace C, as indicated by the following equa-
tion:— AC + B = AB + C.

These investigators overlooked a factor of fundamental importance
in conditioning chemical reactivity, viz., the influence of mass.
The importance of the relative amounts of the reacting substances
in determining the course of a reaction was first clearly recognized
by Wenzel * in 1777. It remained for Berthollet,f however, to

* Lehre von der chemischen Verwandtschaft der Korper.
t Essai de Statique Chimique.
312

HOMOGENEOUS EQUILIBRIUM 313

point out the significance of the views advanced by Wenzel.
His first paper on this subject was published in 1799, while acting
as a scientific adviser to Napoleon on his Egyptian expedition.
Under ordinary conditions sodium carbonate and calcium chloride
react according to the equation,

NasCOs + CaClj = 2NaCl + CaCO8,

the reaction proceeding nearly to completion. Berthollet observed
the deposits of sodium carbonate on the shores of certain saline
lakes in Egypt, and pointed out that this salt is produced by the
reversal of the above reaction, the large excess of sodium chloride
in solution in the water of the lakes conditioning the course of
the reaction.

The German chemist Rose* furnished much additional evidence
in favor of the effect of mass on chemical reactions. He pointed
out that in nature, the silicates, which are among the most stable
compounds known, are undergoing a continual decomposition
under the influence of such relatively weak agents as water and
carbon dioxide. The relatively strong specific affinities of the
atoms of the silicates are overcome by the preponderating masses
of water and carbon dioxide in the atmosphere. In 1862 an
important contribution to our knowledge of the effect of mass on
the course of a chemical reaction was made by Berthelot and Pean
de St. Gilles.f They investigated the formation of esters from
alcohols and acids. The reaction between ethyl alcohol and
acetic acid is represented by the equation

02H5OH + CHsCOOH <=± CH3COOC2H5 + H20.

Starting with equivalent quantities of alcohol and acid, the reac-
tion proceeds until about two-thirds of the reacting substances
have been converted into ester and water. In like manner, if
equivalent quantities of ethyl acetate and water are brought
together, the reaction proceeds in the direction indicated by the
lower arrow, until about one-third of the original substances have
been converted into acid and alcohol. In other words the reac-
tion is reversible, a condition of equilibrium resulting when the

* Pogg. Ann., 94, 481 (1855); 95, 96, 284, 426 (1855).

t Ann. Chim. Phys. [3], 65, 385; 66, 5; 68, 225 (1862-1863).

314

THEORETICAL CHEMISTRY

speeds of the two reactions, indicated by the upper and lower
arrows, become equal. If now a fixed amount of acid is taken,
say 1 equivalent, and the quantity of alcohol is varied, a corre-
sponding displacement of the equilibrium follows.

The following table gives the results obtained by Berthelot and
P6an de St. Gilles for ethyl alcohol and acetic acid. The first
and third columns give the number of equivalents of alcohol to
1 equivalent of acetic acid, and the second and fourth columns
give the percentage of ester formed.

Equivalents
of Alcohol.

Ester
Formed.

Equivalents
of Alcohol.

Ester
Formed.

0.2

19 3

82.8

0.5

42 0

4.0

88.2

1 0

66 5

12.0

93 2

1 5

77.9

50.0

100 0

The effect of increasing the mass of alcohol on the course of the
reaction is very beautifully shown by the above results.

The Law of Mass Action. While the influence of the relative
masses of the reacting substances in conditioning chemical reac-
tions was thus fully established, it was not until 1867 that the law
governing the action of mass was accurately formulated.

In that year Guldberg and Waage,* two Scandinavian investiga-
tors, enunciated the law of mass action as follows: — The rate, or
speed, of a chemical reaction is proportional to the active masses of
the reacting substances present at that time. Guldberg and Waage
defined the term * 'active mass" as the molecular concentration
of the reacting substances. It is to be carefully noted that the
amount of chemical action is not proportional to the actual
masses of the substances present, but rather to the amounts
present in unit volume. The law is generally applicable to
homogeneous systems; that is, to those systems in which ordi-
nary observation fails to reveal the presence of essentially
different parts. The amount of chemical action exerted by a

' Etudes sur les Affinitfe Chimiques, Jour, prakt. Chem. [2], 19, 69 (1879).

HOMOGENEOUS EQUILIBRIUM 315

substance can be determined, either from its effect on the equili-
brium, or from its influence on the speed of reaction.

In order to apply the law of mass action practically, it must be
formulated mathematically. Let a and b denote the molecular
concentrations of the substances initially present in a reversible
reaction. According to the law of mass action, the rate at which
these substances combine is proportional to the active masses
of each constituent, and therefore to their product, ab. The
initial speed of the reaction at the time to is therefore,

Speedy QO a6, or Speed/0 = k • 06,

in which the proportionality factor fc, is known as the velocity
constant. As the reaction proceeds, the molecular concentrations
of the original substances steadily diminish, while the molecular
concentrations of the products of the reaction steadily increase.
Let us assume that after the interval of time t, x equivalents of
the products of the reaction have been formed. The speed of
the original reaction will now be

Speed/ = k (a — x) (b — x).

As the reaction proceeds, the tendency of the products to combine
and reform the original substances increases. At the time t,
when the concentration of the products is x, the speed of the
reverse reaction will be

Speed* = ki • x2,

where fci is the velocity constant of the reverse reaction.

We thus have two reactions proceeding in opposite directions:
the speed of the direct reaction continuously diminishes while
that of the reverse reaction continually increases. It is evident
that a point must ultimately be reached at which the speeds of
the direct and reverse reactions become equal, and a condition
of dynamic equilibrium will be established. Let Xi represent the
value of x when equilibrium is attained; we then have

Speed direct = k (a -Xi) (6 - Xi) = Speed reverse =

(a —Xj) (b — Xj) _ fci _ ~
xi2 ~ k ~ A'

316 THEORETICAL CHEMISTRY

in which K is known as the equilibrium constant. Since the veloc-
ity constants k and k\, are independent of the concentration, it
follows that the above equation holds for all concentrations.
Therefore, if the value of the equilibrium constant of a reaction
is known, the equilibrium conditions can be calculated for any
concentrations of the reacting substances. When more than one
mol of a substance is involved in a reaction, each mol must be
considered separately in the mass action equation.
Thus let

represent any reversible reaction, in which n\ mols of A\ and n2
mols of At react to form n\ mols of A\ and n^ mols of A?,'. When
equilibrium is attained, we shall have

Jii'tfr' I.

C/ A 'V> t I ... /V

CA1CAZ • ' • Al

in which the symbol c is used to denote the active mass or molecular
concentration of the substances involved in the reaction. This
is a perfectly general form of the mass-action equation. Since at
anyone temperature, concentration and pressure are proportional,
we may write equation (1) in the following form:

which, in the case of gaseous equilibria, is often a more convenient
form of the equation.

The relation between the two equilibrium constants, Kc and Kp,

can be easily determined, as follows: — Since c =~ ^ p^T
have, on substituting this valye of c in equation (1),

\RT) \RTt ' • •

BTA

\RT! ' ' '

HOMOGENEOUS EQUILIBRIUM 317

or indicating the sum of the initial number of mols by Snand the
sum of the final number of rnols by Sn', we have
Kf =

It is evident, therefore, that in reactions where the same number
of mols occur on both sides of the equality sign, Kc = K p. Equa-
tion (1) (or equation (2)) is sometimes known as the reaction
isotherm. While the law of action may be proved thermody-
namically, a much simpler kinetic derivation has been given by
van't Hoff. If we assume that the rate of chemical change is pro-
portional to the number of collisions per unit of time between the
molecules of the reacting substances, then in the reaction

niAi + nsA2 + . . . = ni'Ai + n2'A2' + . . . ,
the velocity of the direct change will be fcc^c^, • • • and the
velocity of the reverse reaction will be kic^,c^n,. . . .
At equilibrium, the two velocities will be equal and, therefore,

As a consequence of the assumptions involved in both the thermo-
dynamic and the kinetic proofs of the law of mass action, it fol-
lows that the law is only strictly applicable to very dilute solutions.
Notwithstanding this limitation, experimental results indicate
that it frequently holds for moderately-concentrated solutions.

Equilibrium in Homogeneous Gaseous Systems.

(a) Decomposition of Hydriodic Acid. A typical example of
equilibrium in a gaseous system is afforded by the decomposition
of hydriodic acid, as represented by the equation

Ha + !,<=* 2 HI.

This reaction has been thoroughly investigated by Hautefeuille,

Lemoine and Bodenstein.* The reaction is well adapted for

investigation since it proceeds very slowly at ordinary temper-

* Zeit. Phys. Chem., 22, 1 (1897).

318 THEORETICAL CHEMISTRY

atures, while at the temperature of boiling sulphur, 448° C.,
equilibrium is established quite rapidly. If the mixture of gases
is maintained at 448° C. for some time and is then cooled quickly,
the respective concentrations of the components of the mixture
can be determined by the ordinary methods of chemical analysis.
Various mixtures of the gases are sealed in glass tubes and heated
for a definite time in the vapor of boiling sulphur. The tubes
are then cooled rapidly to the temperature of the room and, after
the iodine and hydriodic acid have been removed by absorption in
potassium hydroxide, the amount of free hydrogen present in each
tube is measured.
Applying the law of mass action to the above equation, we have

CH, • c/2 _ ~
^72 -AC.
^m

Expressing the analytical results in mols, let a mols of iodine be
mixed with b mols of hydrogen, and let 2 x mols of hydriodic acid
be formed. Then when equilibrium is established, a — x will be
the amount of iodine vapor and b — x will be the amount of hydro-
gen present. The concentrations being directly proportional to
the amounts present, we may substitute these values for c//2, c/2,
and CHI in the mass-action equation. The following expression

is thus obtained: —

(b - x) (a - x) „
- -

Solving the equation for x, we obtain

a +b - Va? + b*-ab(2- 16 J

= - __ --

Since, according to Avogadro's law, equal volumes of all gases
contain the same number of molecules, volumes may be sub-
stituted for a, by and x. Bodenstein expressed his results in terms
of volumes reduced to standard conditions of temperature and
pressure. On analyzing equilibrium mixtures, Bodenstein found
that at 448° C., Kc « 0.01984, and at 350° C., Ke = 0.01494.

Having determined the value of the equilibrium constant, he
made use of this value in calculating the volume of hydriodic

HOMOGENEOUS EQUILIBRIUM

319

acid which should be obtained from known volumes of hydrogen
and iodine. A comparison of the calculated and observed values
showed excellent agreement. The following table contains a few
of the results obtained by Bodenstein at 448° C.

Hydrogen,
6.

Iodine,

HI calculated,
2z.

HI observed,
2x.

20.57

5.22

10 19

10 22

20 60

14 45

25 54

25 72

15 75

11 90

20 65

20 70

14.47

38.93

27 77

27 64

8.10

2 94

5 64

5 66

8 07

9 27

13 47

13 34

It is of interest to note that a change in pressure does not
alter the equilibrium in this gaseous system. Making use of the
partial pressures of the components of the gaseous system instead
of the concentrations, we have

PH, • Pi2
PHI

Kp.

Now let the total pressure on the system be increased to n times
its original value; then the partial pressures are all increased in
the same proportion, and we have

which is equivalent to the original expression, since n cancels
out. The equilibrium is thus seen to be independent of the pres-
sure. This is only true for those systems in which a change in
volume does not occur.

(6) Dissociation of Phosphorus Pentachloride. When phos-
phorus pentachloride is vaporized it dissociates according to the
following equation

Applying the law of mass action, we have

320 THEORETICAL CHEMISTRY

Starting with 1 mol of phosphorus pentachloride, which if undis-
sociated would occupy the volume V, under atmospheric pressure,
and letting a denote the degree of dissociation, the molecular con-
centrations at equilibrium will be as follows: —

Letting (1 + a) V = V, and substituting in the above equation,
we have

(1 - a) F

At 250° C. phosphorus pentachloride is dissociated to the extent
of 80 per cent. Under atmospheric pressure 1 mol will be present

273 I 250
in 22.4 — ^ — liters = V. The final volume will, therefore, be

V = (1 + 0.8) (22 A

.8) (22.

The value of the equilibrium constant — usually designated in
cases of dissociation, the dissociation constant — is, therefore,

_ (0.8)'

"-- - - - -

(1 - 0.8) (1 + 0.8) ( 22.4 273+3250)

Having obtained the value of Ke, the direction and extent of the
reaction at 250° C. can be determined, provided the initial molec-
ular concentrations are known. The reaction is accompanied by
a change in volume, and, therefore, the equilibrium is displaced by
a change in pressure. Making use of the partial pressures of the
components of the gaseous mixture, we have

where p\ and p2 are the partial pressures of phosphorus penta-
chloride and the products of the dissociation, phosphorus tri-

HOMOGENEOUS EQUILIBRIUM 321

chloride and chlorine, respectively. Let the total pressure be
increased n-times, then

npi pi

It is apparent from this equation, that the equilibrium is not
independent of the pressure, an increase in pressure being accom-
panied by a diminution of the dissociation. An important point
in connection with dissociation, first observed by Deville,* is the
effect on the equilibrium of the addition of an excess of one of the
products of dissociation. For example, in the equilibrium

an excess of chlorine or of phosphorus trichloride, drives back the
dissociation. If p\ denotes the partial pressure of phosphorus
pentaehloride, pz that of phosphorus trichloride, and p$ that of
chlorine, then we have

Now let an excess of chlorine be added; this will cause the value
of PC to increase. Since the value of Kp is constant, the value of
pz must diminish and that of pi must increase. Hence, the addi-
tion of an excess of either product of dissociation causes a diminu-
tion of the amount of the dissociation.

This is a somewhat more complex gaseous system than either 01
the foregoing systems. When equilibrium is established, let
pL be the partial pressure of the carbon dioxide, p^ the partial pres-
sure of carbon monoxide, and p3 the partial pressure of oxygen,
then we have

- 5 — — Jt«.
Pi2

At 3000° C. and under atmospheric pressure, carbon dioxide is
* Logons sur la dissociation, Paris (1866).

322 THEORETICAL CHEMISTRY

40 per cent dissociated. The partial pressures of each of the com-
ponents may be readily calculated as follows : —

„ - 2(1-0.40)

Pl 2 (1 - 0.40) + 3 X 0.40 '

_ _ 2 X 0.40 _

** 2 (1 - 0.40) + 3 X 0.40 '

°'40 -0.17.

^ 2 (1 - 0.40) + 3 X 0.40
Substituting these values in the above equation, we obtain
_ (0.33)' X 0.17

V -- (050)5

The dissociation constant for carbon dioxide may have a different
value if the equation is written in the form

Applying the law of mass action, we have

Substituting the above values of the partial pressures, we obtain

Kp = 0.272.

Equilibrium in Liquid Systems. The reaction between an
alcohol and an acid to form an ester and water may be taken as
an example of equilibrium in a liquid system. In the reaction

C2H5OH + CH3COOH ^±CH3COOC2H5 + H20,
let a, 6, and c represent the number of mols of alcohol, acid and
water respectively, which are present in V liters of the mixture,
and let x denote the number of mols of ester and water which
have been formed when the system has reached equilibrium.

The active masses of the components will then be,

r - a~~x r - *>- x t r x . , r _ c + x

^alc. -- y > v/acid -- y — > ^ester ~ y > «Jia U water — y •

Applying the law of mass action, we obtain

x (c + x) ~

(a -*)(&-*) "^

HOMOGENEOUS EQUILIBRIUM

323

In this case the value of the equilibrium constant is independent
of the volume. This reaction has been studied, as already men-
tioned, by Berthelot and P£an de St. Gilles.* They found that
when equivalent amounts of alcohol and acid are mixed, the reac-
tion proceeds until two-thirds of the mixture is changed into ester
and water. Hence, we find

Having determined the value of Kc, it may now be used to cal-
culate the equilibrium conditions for any initial concentrations
of the substances involved in the reaction. As an illustration,
we will take 1 mol of acetic acid and treat it with varying amounts
of alcohol, the initial mixture containing neither of the products of
the reaction. The equation takes the form

(a - s) (1 - x)
Solving for x, we have

x = | (1 + a - Va2 - a + 1).

A comparison of the observed and calculated values given in the
accompanying table shows that the agreement is excellent, even
in the more concentrated solutions, where we might reasonably
expect that the mass law would cease to hold.

Alcohol,

Ester
(observed),
x.

Ester
(Calculated),
x.

0.05

0 05

0 049

0 OS

0 078

0 18

0 171

0.171

0 28

0 226

0 232

0 33

0 293

0 311

0.50

0 414

0 523

0 67

0 519

0 528

1.0

0.665

0.667

1 5

0 819

0.785

2.0

0.858

0.845

2 24

0.876

0 864

8 0

0.966

0.945

* Loc, cit.

324 THEORETICAL CHEMISTR^

The Variation of the Equilibrium Constant with Temperature.

van't Hoff showed that the displacement of equilibrium due to
change in temperature is connected with the heat evolved or ab-
sorbed in a chemical reaction by the equations

and

d(log.Kp) -

tSL = 5?" d\

jrp T>Hnz ' \A/

(2)

dT RT2 '

where Qv and Qp are the heats of reaction at Constant volume and
constant pressure respectively, and where R and T have their
usual significance. Equation (1) is known as the reaction isochore.
Both equations show that the rate of change of the natural log-
arithm of the equilibrium constant with temperature is equal to
the total heat of reaction divided by the molecular gas constant
times the square of the absolute temperature at which the
reaction takes place. Equations (1) arid (2) hold only for displace-
ments of the equilibrium due to infinitely small changes in temper-
ature. In order to render these equations applicable to concrete
equilibria, it is necessary to integrate them. The integration of
these expressions can only be performed if Q is constant. For
small intervals of temperature, Q is practically independent of
the temperature, and for larger intervals we may take the value
of Q which corresponds to the mean of the two temperatures
between which the integration is performed. Integrating equa-
tions (1) and (2) on this assumption, we obtain

log.1^ - log. XC| = jr ~ r, (3)

and

(4)

Passing to Briggsian logarithms, and putting R = 1.99 calories,
equations (3) and (4) become

(5)

HOMOGENEOUS EQUILIBRIUM 325

and

(6)

We shall now proceed to show how these important equations
may be applied to several typical, equilibria.

(a) Vaporization of Water. The equilibrium between a liquid
and its vapor is conditioned by the pressure of the vapor,
this in turn being dependent upon the temperature. In this case
of physical equilibrium, we have Kpi = pi, and KP<1 = p2. The
value of Qp for water can be calculated from the following data: —

Ti = 273°, pi = 4.54 mm. of mercury,

T* = 273° + 1T.54, p2 = 10.02 mm. of mercury.

Substituting in equation (6), we have

4.581 (log 10.02 - log 4.54) 273 X 284.5
y" 284.5 - 273

or -QP = -10,670 calories.

The value of Qp obtained by experiment is — 10,854 calories.

(b) Dissociation of Nitrogen Tetroxide. In the reaction

N204 <& 2 NO2,

the following values for the dissociation of N204 have been ob-

tained: —

r1==273°+ 26°.l, «! = 0.1986,

T2 = 273° + 111°.3, «2 = 0.9267.

If the dissociation takes place under a pressure of 1 atmosphere,
then the partial pressures of the component gases will be

Pw' = l-1«~+2«' and P^i-a + 2*-

The values of KPI and KPI are, then, according to the law of
mass action as follows: —

326 THEORETICAL CHEMISTRY

and

Substituting in equation (6) and solving for Qp, we obtain

A Coi fi 4 X (0.9267)2 , 4 X (0.1986)2] OOQ , _
4.58l|log nrjp^-Wf^^

y* 384.3 - 299.1

QP = — 12,260 calories per mol of N2O4.

In a reaction which is accompanied by no thermal change,
Q = 0, and the right-hand side of equations (1) and (2) becomes
equal to zero. In other words, in such a reaction a change in
temperature does not cause a displacement of the equilibrium.

The reaction,

C2H5OH + CH3COOH *± CH3COO.C2H5 + H20,

is accompanied by such a small thermal change that it may be
considered as zero, and according to the above reasoning there
should be only a very slight displacement of the equilibrium when
the temperature is varied. Berthelot found that at 10° C.,
65.2 per cent of the alcohol and acid are changed into ester, and
at 220° C., 66.5 per cent of the mixture is transformed into ester.
As will be seen, an increase of 210° produces hardly any displace-
ment of the equilibrium.

PROBLEMS.

1. When 2.94 mols of iodine and 8.10 mols of hydrogen are heated at
constant volume at 444° C. until equilibrium is established, 5.64 mols
of hydriodic acid are formed. If we start with 5.30 mols of iodine and
7.94 mols of hydrogen, how much hydriodic acid is present at equilibrium
at the same temperature? Ans. 9.49 mols.

2. At 2000° C., and under atmospheric pressure, carbon dioxide is
1.80 per cent dissociated according to the equation

Calculate the equilibrium constant for the above reaction using partial
pressures. Ans. 3 X 10~6.

HOMOGENEOUS EQUILIBRIUM 327

3. What is the equilibrium constant in the preceding problem, if the
concentrations are expressed in mols per liter? Ans. 1.61 X 10~8.

4. When 6.63 mols of amylene and 1 mol of acetic acid are mixed,
0.838 mol of ester is formed in the total volume of 894 liters. How much
ester will be formed when we start with 4.48 rnols of amylene and 1 mol
of acetic acid in the volume of 683 liters? Ans. 0.8111 mol.

5. If 1 mol of acetic acid and 1 mol of ethyl alcohol are mixed, the
reaction

C2H6OH + CHaCOOH ?± CH3COOC2H5 + H20,

proceeds until equilibrium is reached, when -J- mol of ethyl alcohol, J mol
of acetic acid, J mol of ethyl acetate, and f mol of water are present. If
we start (a) with 1 mol of acid and 2 mols of alcohol; (b) with 1 mol of
acid, 1 mol of alcohol, and 1 mol of water; (c) with 1 mol of ester and 3
mols of water, how much ester will be present in each case at equilibrium?
Ans. (a) 0.845 mol, (b) 0.543 mol, (c) 0.465 mol.

6. In the reaction

we find, since J O = I 02, for

__ 9
PHCI ' V p0

the values 3.02 at 386° C. and 2.35 at 419° C. Calculate the heat evolved
by the reaction under constant pressure. Ans. 6827 cal.

7. Above 150° C. N02 begins to dissociate according to the equation
N02^NO+|02.

At 390° C. the vapor density of N02 is 19.57 (H = 1), and at 490° C.
it is 18.04. Calculate the degree of dissociation according to the above
equation at each of these temperatures; the equilibrium constants
expressing the concentrations in mols per liter; and the heat of dissocia-

tion of N02.

Ans. on = 0,35, <*2 = 0.55, Ki = 2.884 X 10~2.

K2 = 7.173 X 10-*. Q « -9407 cal.

CHAPTER XVI.
HETEROGENEOUS EQUILIBRIUM.

Heterogeneous Systems. We have now to consider equilibria
in systems made up of matter in different states of aggregation.
Such systems are termed heterogeneous systems, as distinguished
from those dealt with in the preceding chapter where the compo-
sition is uniform throughout. The physically distinct portions of
matter involved in a heterogeneous system are known as phases,
each phase being homogeneous and separated from the other
phases by definite bounding surfaces. Thus, ice, liquid water
and vapor constitute a physically heterogeneous system. Another
heterogeneous system is formed by calcium carbonate and its
dissociation products, calcium oxide and carbon dioxide. The
equilibrium between a solid, its saturated solution, and vapor
affords an illustration of a still more complex heterogeneous
system.

Application of the Law of Mass Action to Heterogeneous
Equilibria. It has been shown in the preceding chapter that
the law of mass action may be applied to homogeneous equilibria
provided the molecular condition of the reacting substances is
known.

When we attempt to apply the law of mass action to hetero-
geneous equilibria, especially where solids are involved, the
problem presents difficulties. In his investigation of the dis-
sociation of calcium carbonate, according to the equation

Debray * showed that just as every liquid has a definite vapor
pressure corresponding to a certain temperature, so there is a
definite pressure of carbon dioxide over calcium carbonate at a
definite temperature. Furthermore, the pressure was found to
be independent of the amount of calcium carbonate present.

* Compt. rend., 64, 603 (1867).
328

HETEROGENEOUS EQUILIBRIUM 329

Guldberg and Waage * showed that the law of mass action can
be applied to such heterogeneous equilibria, provided that the
active masses of the solids present are considered as constant.

Nernst pointed out that this statement of Guldberg and Waage
can be easily reconciled with experimental facts. In a hetero-
geneous equilibrium involving solids, it is only necessary to con-
sider the gaseous phase, the active mass of a solid being equivalent
to its concentration in the gaseous phase. That is, every solid
is to be looked upon as possessing, at a definite temperature, a defi-
nite vapor pressure which is entirely independent of the amount of
solid present. Such substances as arsenic, antimony, and cadmium
are known to have appreciable vapor pressures at relatively low
temperatures, and it is quite reasonable to suppose that every
solid substance exerts a definite vapor pressure at a definite temper-
ature, even though we have no method sufficiently refined to meas-
ure such minute pressures.

Since the active mass of a solid remains constant so long as
any of it is present, the application of the law of mass action
to certain heterogeneous equilibria is, in general, simpler than
its application to homogeneous systems. The truth of this state-
ment will be evident after a few typical heterogeneous systems
have been considered.

(a) Dissociation of Calcium Carbonate. In the reaction

let TI and 7r2 represent the pressures due to the vapor of calcium
carbonate and calcium oxide respectively, and let p denote the
pressure of the carbon dioxide. Applying the law of mass action,
we obtain

But since TI and x2 are constant at any one temperature, the
equation becomes

P - K9;

or, the equilibrium constant at any one temperature is solely
dependent upon the pressure of the carbon dioxide evolved. The

* Loc. cit»

330

THEORETICAL CHEMISTRY

accompanying table gives the values of the pressure of carbon
dioxide corresponding to various temperatures.

Temperature,
Degrees.

Pressure in
Millimeters of
Mercury.

547

610

625

740

255

745

289

810

678

812

753

865

1333

(b) Dissociation of Ammonium Hydrosulphide. When solid
ammonium hydrosulphide is heated, it is almost completely dis-
sociated into ammonia and hydrogen sulphide as shown by the
following equation: —

[NH4HS]^(NH3)

This reaction was investigated by Isambert,* who found that the
total gas pressure at 25°. 1 C. is equal to 501 mm. of mercury.
Since the partial pressures of the ammonia and hydrogen sulphide
are necessarily the same, each must be approximately equal to
250.5 mm., the relatively small pressure due to the undissociated
vapor of the ammonium hydrosulphide being neglected. Let IT
be the partial pressure of the vapor of ammonium hydrosulphide,
and let pi and p^ be the partial pressures of the ammonia and
hydrogen sulphide. Applying the law of mass action, we have

Pi'p* _ v m

- — i^p. {i)

Since ?r is constant at any one temperature, equation (1) becomes

pi • p2 = KP'.
According to Dalton's law of partial pressures, we have

P = Pi + Pz + *,
* Compt. rend., 93, 595, 730 (1881).

HETEROGENEOUS EQUILIBRIUM

331

where P is the total pressure. Neglecting the relatively small
pressure ?r, we may write

P = Pi + ft.
HencCj since pi = p%,

2 = Pi = Pa-
Substituting these values in equation (1), we obtain

P2 _ , _ (501
T-AP --T

^ = IT.' = Ml! = 62,750.

The value of the equilibrium constant may be checked by observ-
ing the effect on the system of the addition of an excess of either
one of the products of the dissociation. The accompanying table
gives the results of a few of Isambert's experiments.

Pressure of
Ammonia.

Pressure of
Hydrogen
Sulphide.

PNH3-PH2S = Kp'.

208
138
417
453

294
458
146
143

61,152
63,204
60,882
64,779

Mean 62,504

As will be seen, the mean value of the equilibrium constant agrees
well with the value found for equivalent amounts of the products
of dissociation.

(c) Dissociation of Ammonium Carbamate. The dissociation of
ammonium carbamate takes place according to the equation
ONH4

OC/
X

NH2

This dissociation has been investigated by Horstmann.* Applying
the law of mass action, we have

'» (2)

* Lieb. Ann., 187, 48 (1877).

332 THEORETICAL CHEMISTRY

where pi and p2 are the partial pressures of ammonia and carbon
dioxide respactively, and where TT is the partial pressure of ammo-
nium carbamate. Since T is constant, equation (2) becomes

If P denotes the total gaseous pressure, and TT is neglected as in
the preceding example, we have, since three mols of gas are
formed

9 4P2 , P

Pi? = — and p2 = -3 •

Substituting these values in equation (2), we have

27 ~j"

This equation has also been tested by Isambert * by adding an
excess of ammonia or carbon dioxide to the dissociating system.
He found that the value of the equilibrium constant remains
practically constant. The addition of a foreign gas was shown
to be without effect on the dissociation.

(d) Dissociation of the Hydrates of Copper Sulphate. Many inter-
esting examples of heterogeneous equilibrium are furnished by hy-
drated salts. Thus, if crystallized copper sulphate, CuS04.5 H20,
is placed in a desiccator, it gradually loses water of crystallization
and ultimately only the anhydrous salt remains. If the desiccator
be provided with a manometer and is so arranged that the tem-
perature can be maintained constant, it is possible to observe the
changes in vapor pressure accompanying the process of dehydra-
tion. At the temperature of 50° C., the pressure over completely
hydrated copper sulphate is found to remain constant at 47 mm.
until the salt has been deprived of two molecules of water, when
it drops abruptly to 30 mm. and remains constant until two more
molecules of water have been lost. It then drops again to 4.4 mm.
and remains constant until dehydration is complete.

The successive stages of the dehydration are shown in the accom-
panying diagram, Fig. 76. The constant pressures observed in
the dehydration correspond to the successive equilibria involved.

* Loc. cit.

HETEROGENEOUS EQUILIBRIUM

333

At 50° C. the pentahydrate and the trihydrate are in equilibrium,
a pressure of 47 mm. being maintained so long as any of the penta-
hydrate is present. When all of the pentahydrate is used up,
then the trihydrate begins to undergo dehydration into the
monohydrate. This is a new equilibrium and the pressure of the

47mm

80 mm

6H2O

3HSO
Composition

Fig. 76.

OH2O

aqueous vapor necessarily changes, and remains constant so long
as any trihydrate remains. The last stage corresponds to the
equilibrium between the monohydrate and the anhydrous salt.
The following equations represent the three successive equilibria: —

(1) CuS04 • 5 H20 ^ CuS04 • 3 H20 + 2 H20,

(2) CuS04 • 3 H20 *± CuS04 • H2O + 2 H2O,
(3)

Applying the law of mass action to the first of the above equi-
libria, we have

tr\

in which ir\ and T2 denote the partial pressures due to the hydrates
CuS04.5 H2O and CuSOi.3 H2O respectively, and p denotes the

334

THEORETICAL CHEMISTRY

pressure of aqueous vapor. Since TTI and 7r2 are constant, the
above expression simplifies to the following

In a similar manner it may be shown that the pressure of aqueous
vapor in the other equilibria must be constant. It must be
clearly understood that the observed pressure is only definite
and fixed when two hydrates are present. If the dehydration

Tee

Temperature

Fig. 77.

were conducted at another temperature than 50° C. the equilibrium
pressure would be different. The vapor pressure curves of the
different hydrates are shown in the temperature-pressure diagram
of Fig. 77.

Heat of Dissociation of Solids. When the products of the
dissociation of a solid are gaseous, it has been pointed out by De
Forcrand * that the ratio of the heat of dissociation of 1 mol of

* Ann. Chim. Phys. [7L 28, 545.

HETEROGENEOUS EQUILIBRIUM

335

solid to the absolute temperature at which the dissociation pres-
sure is equal to 1 atmosphere, is constant. Or, denoting the heat
of dissociation by Q and the absolute temperature by Ty De For-
crand's relation may be expressed thus,

~ = constant = 33.

Nemst has shown that the value of the constant in this relation
is not independent of the temperature. Thus, the value of the
ratio at 100° C. is 29.7, while at 1000° C. it is 37.7. Up to the
present time no expression has been derived in which the variation
of the ratio with the temperature is included.

Distribution of a Solute between Two Immiscible Solvents.
When an aqueous solution of succinic acid is shaken with ether,
the acid distributes itself between the ether and the water in such
a way that the ratio between the two concentrations is always
constant. It will be seen that the distribution of the succinic
acid between the two solvents is analogous to that of a substance
between the liquid and gaseous phases (see page 170), and there-
fore the laws governing the latter equilibrium should apply equally
to the former. Nernst * has shown that (a) // the molecular
weight of the solute is the same in both solvents, the ratio in which it
distributes itself between them is constant at constant temperature,
or in other words, Henry's law is applicable; and (b) If there are
several solutes in solution the distribution of each solute is the same as
if it were present alone. This is clearly Dalton's law of partial
pressures. The ratio in which the solute distributes itself between
the two solvents is termed the coefficient of distribution or partition.
The following table gives the results of three experiments on the
distribution of succinic acid between ether and water.

Concentration
in Water.

Concentration
in Ether.

Distribution
Coefficient.

43.4

43. 8
47.4

7.1
7.4
7.9

6.1
5.9
6.0

Zeit. phys. Chem., 8, 110 (1891).

336

THEORETICAL CHEMISTRY

As will be seen the distribution coefficient is constant, show-
ing that Henry's law applies. When the molecular weight of a
solute is not the same in both solvents the distribution coefficient
is not constant, and conversely, if the distribution coefficient is
not constant, we infer that the molecular weights of the solute
in the two solvents are not identical.

Let us assume that a solute whose normal molecular weight is
A, when shaken with two immiscible solvents undergoes polymeri-
zation in one of them, its molecular weight being An. We then
have the equilibrium

applying the law of mass action, we have

CAn rr

— = Ac.

CA«

If the molecular weight in one solvent is twice the molecular
weight in the other, then n = 2, and

Ciz

— or

r= = constant.

Thus Nernst found the following concentrations of benzoic acid
when it was shaken with benzene and water.

c, (Water).

c2 (Benzene).

C|.

0 0150
0.0195
0.0289

0 242
0.412
0 970

0.062
0 048
0.030

0 0305
0 0304
0 0293

As will be seen, the values of the ratio c\/c^ steadily decrease,
while on the other hand, the values of the ratio Ci/V^ remain
constant. This shows, therefore, that benzoic acid has twice the
normal molecular weight in benzene.

The Solution of a Solid in a Non-dissociating Solvent When
a solid is brought in contact with a non-dissociating solvent, it
continues to dissolve until the solution becomes saturated. A
condition of equilibrium then obtains, the rates of solution and

HETEROGENEOUS EQUILIBRIUM 337

precipitation being the same. This is plainly a case of hetero-
geneous equilibrium. If c is the concentration of the dissolved
substance, and TT is the concentration of the undissolved solid, then
according to the law of mass action

--K

or since T is constant,

c = Kc'.

Variation of the Constant of Heterogeneous Equilibrium with
Temperature. The reaction isochore equation of van't Hoff

which has been shown to connect the displacement of a homo-
geneous equilibrium with change in temperature, applies equally
well to heterogeneous equilibria. The following examples will
serve to illustrate its application in such cases.

(a) Dissociation of Ammonium Hydrosulphide. In the reaction
representing the dissociation of ammonium hydrosulphide,

let pi and p2 be the partial pressures of ammonia and hydrogen
sulphide, and let w be the partial pressure of ammonium hydro-
sulphide. Then as has been shown (see page 331),

Jf/-T,

where P is the total gaseous pressure. From the following data : —

Ti = 273° + 9°.5, Pi = 175 mm. of mercury,
and

Tz = 273° + 25°. 1, P2 = 501 mm. of mercury,

we have, on applying the reaction isochore equation, and solving
forQp,

298.1

" 298.1 - 282.5

or Qp = - 22,740 calories.

338 THEORETICAL CHEMISTRY

This result agrees well with the value, -22,800 calories, found
by direct experiment.

(b) Solution of Sucdnic Add. The concentration of succinic
acid (in a saturated solution) and the temperature, are the factors
which determine the equilibrium in this case. In the equation

d(\OgeKc) = -Qv

dT RT2 '

Kc = c, where c is the concentration of succinic acid in a saturated
solution. The following experimental data, due to van't Hoff,
enables us to calculate the heat of solution of the acid.

T! = 273° c = 2.88 grams per 100 grams of water,

and

T2 = 273° + 8°.5, c = 4.22 grams per 100 grams of water.

Substituting in the reaction isochore equation and solving for Qc,
we have

_0 4.581 (log 4.22 - log 2.88) 273^X 281.5

y" " 281.5 - 273

Qv = - 6871 calories.

The value of the heat of solution for 1 mol of succinic acid as
found by direct experiment is —6700 calories.

The Phase Rule. While it is possible to apply the law of
mass action to certain heterogeneous equilibria there are numerous
cases where its application is either difficult or impossible. To
deal with such heterogeneous systems we make use of a general-
ization discovered by J. Willard Gibbs,* late professor of mathe-
matical physics in Yale University. This generalization was first
stated by Gibbs in 1874, and is commonly known as the phase rtile.
Before entering upon a discussion of the phase rule, it will be
necessary to define a few of the terms employed.

The composition of a system is determined by the number of
independent variables or components involved. Thus in the
system — ice, water, and vapor — there is but a single com-
ponent. In the system

Trans. Connecticut Academy, Vols. II and III, 1875-8.

HETEROGENEOUS EQUILIBRIUM 339

while there are three constituents of the equilibrium, only two of
these need, be considered as components, for the amount of any
one constituent is not independent of the amounts of the other
two, as the following equations show: —

CaO + CO2 = CaCO3,
CaC03 - CaO = C02,
CaC03 - CO2 = CaO.

In general, the components are chosen from the smallest number
of independently-variable constituents required to express the
composition of each phase entering into the equilibrium, even
negative quantities of the components being permissible.

The number of variable factors, — temperature, pressure, and
concentration, — of the components which must be arbitrarily fixed
in order to define the condition of the system, is known as the
degree of freedom of the system. For example, a gas has two
degrees of freedom since two of the variables, temperature, pres-
sure or volume, must be fixed in order, to define it; a liquid and its
vapor has only one degree of freedom, since for equilibrium at a
certain temperature, there can be but a single pressure; and in a
system consisting of a substance in the three states of aggregation
equilibrium can only exist at a single temperature and pressure.

Derivation of the Phase Rule. The following derivation of
the phase rule is due to Nernst. Let us assume a complete hetero-
geneous equilibrium made up of y phases of n components, and let
us fix our attention upon one single phase. This phase will con-
tain a certain amount of each one of the n components, the con-
centrations of which may be designated by Ci, C2, c3, . . . cn.
Since we have assumed complete equilibrium to exist, the slightest
change in concentration, temperature or pressure will alter the
composition of this phase.

This may be expressed by the equation

/ (ci, c2, CB, . . . cn, p, T) = 0,

where / is any function of the variables. Since any change in
one phase implies a corresponding change in the remaining y — 1
phases, it follows that the composition of all the phases is a certain
determined function of the same variables.

340 THEORETICAL CHEMISTRY

The above equation is, then, of the form ascribed to each sepa-
rate phase, and since there are y phases we have y separate equa-
tions. There are, however, n + 2 variables in each equation, so
that if y = n + 2, that is if we have two more phases than com-
ponents, each unknown quantity has a definite known value.
In this case there is only one value for ci, C2, c3, c4, . . . cn, p and
T at which the system can be in equilibrium. Hence when n
components are present in n + 2 phases, we have equilibrium only
for a certain temperature, a certain pressure, and a certain ratio
of concentrations of the single phases. That is, n + 2 phases of n
substances can only exist at a certain point in a coordinate
system. This point is termed the transition point. If one value
be altered then one phase vanishes, and there remain n + 1 phases
of n components, and the problem becomes indeterminate. Thus
it is proved that n components are necessary in order that a system
containing n + 1 phases may exist in complete equilibrium.

The phase rule may be stated as follows: — A system made up
of n components in n + 2 phases can only exist when pressure,
temperature and concentration have definite fixed values; a system
of n components in n + 1 phases can exist only so long as one of the
factors varies; and a system of n components in n phases can exist
only so long as two of the factors vary. If P denotes the number of
phases, C the number of components, and F the number of degrees
of freedom, then the phase rule may be conveniently summarized
by the expression,

Equilibrium in the System, Water, Ice, and Vapor. In this
system we may have one, two, or three phases present, according
to the conditions. Under ordinary circumstances of temper-
ature and pressure, water and water vapor are in equilibrium.
The vapor pressure curve of water is represented by the line OA
in the pressure-temperature diagram (Fig. 78). It is only at
points on this curve that water and its vapor are in equilibrium
Thus, if the pressure be reduced below that corresponding to any
point on OA, all of the water will be vaporized; if, on the other
hand, the pressure be raised above the curve, all of the vapor will

HETEROGENEOUS EQUILIBRIUM

341

ultimately condense to the liquid state. When the temperature
is reduced below 0° C., only ice and vapor are present, the curve
OC representing the equilibrium between these two phases. It is
to be observed that the curve OC is not continuous with OA. At

0:0075
Temperature
Fig. 78.

the point 0, where the two curves intersect, ice, water, and water
vapor are in equilibrium. At this point ice and water must have
the same vapor pressure, otherwise distillation of vapor from the
phase having the higher vapor pressure to that with the lower
vapor pressure would occur, and eventually the phase having the
higher vapor pressure would disappear. This result would be
in contradiction to the experimentally-determined fact that
both solid and liquid phases are in equilibrium at the point 0.
The temperature at which ice and water are in equilibrium with
their vapor under atmospheric pressure is 0° C. Since increase
of pressure lowers the freezing-point of water, the point 0, repre-
senting the equilibrium between ice and water under the pressure
of their own vapor, viz., 4.57 mm., must be a little above 0° C.
The exact temperature corresponding to the point 0 has been
found to be O.°0075 C.

342 THEORETICAL CHEMISTRY

The change in the melting-point of ice due to increasing pressure
is represented by the line OB. This line is inclined toward the
vertical axis because the melting-point of ice is lowered by in-
creased pressure. The point 0 is called a triple point because
there, and there only, three phases are in equilibrium. As is well
known, water does not always freeze exactly at 0° C. If the
containing vessel is perfectly clean, and care is taken to exclude
dust, it is possible to supercool water several degrees below its
freezing-point and measure its vapor pressure.

The dotted curve OA', which is a continuation of OA, represents
the vapor pressure of supercooled water. It will be noticed that
(1) there is no break in the vapor-pressure curve so long as the
solid phase does not separate, and (2) the vapor pressure of super-
cooled water, which is an unstable phase, is greater than that of
ice, the stable phase, at that temperature.

We now proceed to apply the phase rule to this system. In the
formula, C - P + 2 = F, C = 1. It is evident that if P = 3,
then F = 0; or the system has no degree of freedom. We have
seen that the triple point 0 represents such a condition. At this
point ice, water, and water vapor are co-existent, and if either
one of the variables, temperature or pressure, is altered, one of
the phases disappears; in other words, the system has no degree
of freedom. Such a system is said to be non-variant. If in the
above formula, P = 2, then F = 1, and the system has one degree
of freedom, or is univariant. Any point on any one of the curves
OAj OBj or OC represents a univariant system. Take, for exam-
ple, a point on the curve OA . In this case the temperature may
be altered without altering the number of phases in equilibrium.
If the temperature is raised, a corresponding increase in vapor
pressure follows and the system will adjust itself to some other
point on the curve OA. In like manner, the pressure may be
altered without causing the disappearance of one of the phases.
If, however, the temperature is maintained constant, then a change
in the pressure will cause either condensation of water vapor or
vaporization of liquid water. Under these conditions the system
has only one degree of freedom. Again, if P = 1, then F =; 2,
and the system is bivariant, or has two degrees of freedom. The

HETEROGENEOUS EQUILIBRIUM 343

areas included between the curves in the diagram are examples
of bi variant systems. Consider the vapor phase; the temperature
may be fixed at any desired value within the vapor area AOC.
and the pressure may be altered along a line parallel to the vertical
axis without causing a change in the number of phases, provided
the curves OA and OC are not intersected.

The System, Sulphur (Rhombic, Monoclinic), Liquid and
Vapor. This system is more complicated than the preceding
one-component system, since there are two solid phases in addition
to the liquid and vapor phases. At ordinary temperatures, rhom-
bic sulphur is the stable modification. When this is heated
rapidly it melts at 115° C., but if it is maintained in the neighbor-
hood of 100° C. it gradually changes into monoclinic sulphur
which melts at 120° C. Monoclinic sulphur can be kept indefi-
nitely at 100° C. without undergoing* change into the rhombic
modification, or in other words it is the stable phase at this temper-
ature.

It is evident, therefore, that there must be a temperature above
which monoclinic sulphur is the stable form and below which
rhombic sulphur is the stable modification. This temperature
at which both rhombic and monoclinic modifications are in equi-
librium with each other and with their vapor, is termed the
transition point. Its value has been determined to be 95°. 6 C.
The change from one form into the other is relatively slow, so
that it is possible to measure the vapor pressure of rhombic sul-
phur up to its melting-point, and that of monoclinic sulphur
below its transition point. The vapor pressure of solid sulphur,
although very small, has been measured as low as 50° C.

The complete pressure-temperature diagram for sulphur is
shown in Fig. 79. At the point 0, rhombic and monoclinic sul-
phur are in equilibrium with sulphur vapor, this being a triple
point analogous to the point 0 in Fig. 78. The vapor pressure
curves of rhombic and monoclinic sulphur are represented by OB
and OA respectively. The dotted curve OA' which is a continu-
ation of OA is the vapor-pressure curve of monoclinic sulphur in
a metastable region. In like manner OB' represents the vapor-
pressure curve of rhombic sulphur in the metastable condition, B'

344

THEORETICAL CHEMISTRY

being a metastable melting-point. As in the pressure-temper-
ature diagram for water, the metastable phases have the higher
vapor pressures. The effect of increasing pressure on the transi-
tion point 0, is represented by the line OC. This is termed a

Bhombic Sulphur

Vapot

Temperature
Fig. 79.

transition curve, and, since increase in pressure raises the transi-
tion point, the line slopes away from the vertical axis. The
effect of increased pressure on the melting-point of monoclinic
sulphur is shown by the curve AC.

This also slopes away from the vertical axis, but the change in
the melting-point of monoclinic sulphur produced by a given
change in pressure being less than the corresponding change in the

HETEROGENEOUS EQUILIBRIUM 345

transition point, the two curves, OC and AC, intersect at the
point C. The point C corresponds to a temperature of 131° C.
and a pressure of 400 atmospheres. The vapor-pressure curve
of stable liquid sulphur is represented by the curve AD. The
vapor-pressure curve of the metastable liquid phase is represented
by the curve AB' which is continuous with AD. The diagram is
completed by the curve J5'C which represents the effect of pressure
on the metastable melting-point of rhombic sulphur. Mono-
clinic sulphur does not exist above the point C; hence when
liquid sulphur is allowed to solidify at pressures exceeding 400 at-
mospheres, the rhombic modification is formed, whereas under
ordinary pressures the monoclinic modification appears first.

The phase rule enables us to state the exact conditions required
for equilibrium in this system and to check the results of exper-
iment. Thus, according to the formula, C — P + 2 = JF, since
C = 1, the system will be non-variant when P = 3. Since there
are four phases involved, theoretically any three of these may
be co-existent and four triple points are possible. The theoreti-
cally-possible triple points are as follows: —

(2) Rhombic sulphur, monoclinic sulphur, and vapor (0);

(2) Rhombic sulphur, monoclinic sulphur, and liquid (C7);

(3) Rhombic sulphur, liquid, and vapor (J30;

(4) Monclinic sulphur, liquid and vapor (A).

In this particular system all of the four possible triple points can
be realized experimentally. That this is the case is due to the
comparative slowness of the change from rhombic to monoclinic
sulphur above the triple point. If this change were rapid it is
evident that all of the theoretically-possible non-variant systems
could not be realized experimentally.

As in the case of water, the curves in the diagram represent
univariant systems and the areas bivariant systems. The student
is advised to tabulate the univariant and bivariant systems repre-
sented in the pressure-temperature diagram for sulphur.

Two-component Systems. Turning now to two-component
systems we are confronted with a more difficult problem, and one
which includes many special cases. Thus, we may have cases of

346

THEORETICAL CHEMISTRY

anhydrous salts and water, hydrated salts and water, volatile
solutes, two liquid phases, consolute liquids, and solid solutions.
To enter upon a discussion of these would not be profitable, since
they only serve to give greater emphasis to the general truth of
the phase rule. We shall select a few typical two-component
systems for consideration here.

(a) Anhydrous Salt and Water. In the equilibrium diagram
of water (here represented by dotted lines, Fig. 80), we desig-

T Temperature

Fig. 80.

nate the triple point by 0. At this point ice and water have the
same vapor pressure. Similarly, a solution at its freezing-point
has the same vapor pressure as the ice which separates. The
intersection of the vapor-pressure curve for ice, OB, and the vapor-
pressure curve of the solution of the anhydrous salt, 0"A", deter-
mines a new triple point 0". Since the presence of the dissolved
salt tends to diminish the vapor pressure of water, the curve 0" A "
is situated below the curve OA, and for the same reason the triple
point 0" is found to the left of 0. If now we keep an excess of
dissolved substance continually present, all of the liquid phases

HETEROGENEOUS EQUILIBRIUM 347

which are formed will of necessity be saturated solutions. When
these solutions finally freeze they will furnish, not pure ice, but a
mixture of ice and solid salt, known as a cryohydrat'e. By a
partial freezing we can therefore obtain the system: Solid salt,
ice, saturated solution and vapor, or in other words, a system of
n + 2 phases of which the existence is only possible at the freez-
ing temperature Tf of the saturated solution, and under the pres-
sure pf corresponding to the vapor pressure of ice and the saturated
solution. These conditions are represented in the diagram by
the quadruple point 0'. If now we pass from the point 0', increas-
ing the temperature and pressure as prescribed by the curve
O'A', the ice disappears, while the salt, the saturated solution,
and the vapor furnish a series of 3-phase systems. Again start-
ing from the point 0' and lowering the temperature and the pres-
sure as indicated by the curve O'B, the liquid phase disappears,
while the solid salt, ice, and vapor constitute another series of
3-phase systems. This, of course, is on the supposition that the
vapor pressure of the solid salt is negligible. Finally, a consider-
able increase in pressure causes a slight lowering of the temper-
ature corresponding to the quadruple point, the conditions being
represented by the curve O'C'.

All possible non-saturated solutions of the salt will be repre-
sented by points within the area, AOO'A'. Thus, let 0" A11 repre-
sent the vapor-pressure curve of a dilute solution of the salt in
water. The freezing-point of this solution is represented by the
point 0", while 0"C" represents the variation of the freezing-point
of the solution with pressure.

The following table summarizes the possibilities indicated by
the phase rule: —

4 phases; salt, ice, saturated solution, vapor (point, 0');

3 phases; salt, saturated solution, vapor (curve, O'A');

3 phases; salt, ice, vapor (curve, 0'J3);

3 phases; salt, ice, saturated solution (curve, O'C');

2 phases; salt, saturated solution (area, C'OrAf)\

2 phases; salt, water vapor (area, BO'A');

2 phases; salt, ice (area, 50'C');

348

THEORETICAL CHEMISTRY

3 phases; ice, non-saturated solution, vapor (curve, 00');
2 phases; non-saturated solution, vapor ) . . nn'A '\

1 phase; non-saturated solution > ' '*

2 phases; non-saturated solution, ice > , cnnfrf\
1 phase; non-saturated solution S '

As will be seen, there is only one non-variant point in the entire
diagram, viz., the point 0'. In this system there are three degrees
of freedom, since in addition to temperature and pressure the con-
centration of the solution may also be varied.

The pressure-temperature diagram (Fig. 80) having been dis-
cussed, we now turn to the concentration-temperature diagram
for the same system, Fig. 81. In this diagram the abscissae

Temperature
Fig. 81.

represent temperatures and the ordinates, concentrations. For
convenience, corresponding points in Figs. 80 and 81 will be desig-
nated by the same letters. The equilibrium between ice, water
and water vapor is represented by the point 0. If now a small

HETEROGENEOUS EQUILIBRIUM 349

amount of anhydrous salt be added to the water, the freezing-point
will be lowered to 0". As the proportion of salt is increased the
temperature of equilibrium is lowered along the curve 00"0'. A
point is ultimately reached at which the solution becomes saturated,
and on further addition of salt it is not dissolved, but remains in
contact with the ice and saturated solution. This is the cryo-
hydric point, and represents the lowest temperature which can be
obtained in this particular system. The diagram is completed
by the solubility curve of the salt, O'A'. Each point on this
curve represents the concentration of the saturated solution at
all temperatures, from the critical temperature of the solution to
the cryohydric temperature. The meaning of the concentration-
temperature diagram may be made clearer by a consideration of
the behavior of a solution when gradually cooled. Let a repre-
sent a dilute solution of the anhydrous salt. On lowering the
temperature along ab, no change will occur until the curve 00'
is reached; then ice will begin to separate and as the cooling is
continued, the composition of the solution will change along 00'
until it reaches the cryohydric point 0'. Here both salt and ice
will separate, and the solution will solidify completely at the
temperature corresponding to the point 0'. In like manner, if
we start with a concentrated solution represented by the point c
and cool along cd no change will take place until the curve O'A '
is reached; then solid salt will separate and the composition of
the solution will alter along O'A' until the temperature is reduced
to that corresponding to the cryohydric point, when the whole
solution will solidify as in the previous case. This phenomenon
was first systematically investigated by Guthrie * who concluded
that such mixtures of constant composition and definite melting-
point are chemical compounds, and, therefore, he proposed to call
them cryohydrates. It has since been shown that cryohydrates
are not definite chemical compounds. Among the various reasons
which have been advanced to prove the incorrectness of Guthrie's
views, the following are the most cogent: — (1) the physical
properties of a cryohydrate are the mean of the corresponding
properties of the constituents, this being rarely true of chemical

* Phil. Mag. [4J, 49, 1 (1875); [5], i, 49 and 2, 211 (1876).

350

THEORETICAL CHEMISTRY

compounds; (2) the lack of homogeneity of a cryohydrate can
be detected under the microscope; and (3) the constituents are
seldom present in simple molecular proportions.

Applying the phase rule to the above two-component system,
it is evident that there is but one non- variant system: this is
represented by the point 0'. When three phases are co-existent
the system is univariant, when only two phases are present the
system is bivariant, and finally, when only one phase is present
the system acquires three degrees of freedom or is trivariant.
It is evident that a system having three degrees of freedom cannot
be completely represented by a diagram in a single plane. It is
possible, however, to construct a three-dimensional model which
will represent the equilibrium very satisfactorily. Such a model is

Fig. 82.

I. Unsaturated Solution.

II. Salt and Saturated Solution.

III. Ice and Unsaturated Solution,

IV. Ice and Cryohydrate.
V. Salt and Cryohydrate.

shown in Fig. 82, the lettering being made to correspond with
that of the two diagrams, Figs, 80 and 81, from which it is derived.

HETEROGENEOUS EQUILIBRIUM

351

(b) Hydrated Salt and Water. An interesting example is fur-
nished by the system — ferric chloride and water. This system
has been very carefully investigated by Roozeboom.* The con-
centration-temperature diagram, plotted from Roozeboom's data,

0° Temperature -

Fig. 83.

is given in Fig. 83. The freezing-point of pure water is repre-
sented by Ay and the lowering of the freezing-point produced by
the addition of ferric chloride is indicated by the curve AB. At
the cryohydric temperature, — 55° C., ice, Fe2Cl6 • 12 H20, sat-
urated solution, and vapor are in equilibrium, and the system is
non-variant. On adding more ferric chloride, the ice phase dis-
appears, and the univariant system, Fe2Cl6 • 12 H20, saturated
solution, and vapor results. The equilibrium is represented by the
curve BC which may be regarded as the solubility curve of the
dodecahydrate. On continuing the addition of ferric chloride,
the temperature continues to rise until the point C is reached.
Here the composition of the solution is identical with that of the
dodecahydrate, and, therefore, the temperature corresponding to
this point, 37° C., may be looked upon as the melting-point of
Fe2Cl6 • 12 H20. Further addition of ferric chloride will naturally

* Zeit. phys. Chem., 4, 31 (1889); 10, 477 (1892).

352 THEORETICAL CHEMISTRY

lower the melting-point and the equilibrium will alter along the
curve CD. It is thus possible to have two saturated solutions,
one of which contains more water and the other less, than the
hydrate which is in equilibrium with the solution. These solu-
tions are both stable throughout and are nowhere supersaturated.
Roozeboom was the first investigator to discover a saturated
solution containing less water than the solid hydrate with which
it is in equilibrium. This discovery led him to define supersatu-
ration as follows: — "A solution is supersaturated with respect
to a solid phase at a given temperature if its composition is between
that of the solid phase and the saturated solution." At the point
D the curve reaches another minimum which is analogous to the
point J5, except that the heptahydrate, Fe2Cle • 7 H20, takes the
place of ice. Here we have equilibrium between the dodccahydrate,
the heptahydrate, saturated solution, and vapor, and the system
is non-variant. On further addition of ferric chloride another
maximum is reached at E, corresponding to the melting-point of
the heptahydrate. In a similar manner, two other maxima at
greater concentrations of ferric chloride reveal the existence of the
hydrates, Fe2Cl6 • 5 H20, and Fe2Cl6 • 4 H20.

At the three remaining quadruple points the following phases
are in equilibrium : — At F, Fe2Cl6 • 7 H20, Fe2Cl6 • 5 H20, saturated
solution and vapor; at H, Fe2Cl6 • 5 H20, Fe2Cl6 • 4 H20, saturated
solution and vapor; and at K, Fe2Cl6 • 4 H20, Fe2Cl6, saturated
solution and vapor. The solubility of the anhydrous salt is
represented by the curve KL. Metastable solubility and melting-
point curves are represented by dotted lines.

The student should apply the phase rule to this system. If a
fairly dilute solution of ferric chloride is evaporated at 31° C., the
water gradually disappears and a residue of the dodecahydrate
remains. This residue then liquefies and again dries down, the
composition of the residue corresponding to the heptahydrate:
on further standing the phenomenon is repeated, the final and
permanent residue having a composition corresponding to the
pentahydrate. The dotted line ab shows the isothermal along
which the composition varies. It would have been a difficult
matter to explain the alternations of moisture and dryness ob-

HETEROGENEOUS EQUILIBRIUM 353

served in this experiment without the concentration-temperature
diagram.

Alloys. Among the most interesting two-component systems
known are those involving mixtures of metals, or alloys. These
have been made the subject of systematic investigations by num-
erous experimenters among whom may be mentioned Roberts-
Austen, Charpy, Roozeboom, and Heycock and Neville. We have
space to consider only two comparatively-simple cases.

(a) Alloys of Silver and Copper. The conditions of equilibrium
in this binary system have been studied by Heycock and Neville.*
The two components, silver and copper, are not miscible in the
solid state and do not combine chemically. To determine the
curves of equilibrium, mixtures of the two metals in varying pro-
portions were fused and then allowed to cool slowly, the rate of
cooling being observed with a thermocouple, one junction of which
was maintained at constant temperature, while the other junction
was placed in the mixture of molten metals. The terminals of
the thermocouple were connected to a sensitive galvanometer
graduated to read directly in degrees, and the rate of cooling
was followed by the movement of the needle of the galvanometer.
As the mixture cooled, two "breaks" were observed; the first of
these varied with the composition of the mixture, while the second
remained practically constant at 777° C. When the temperatures
corresponding to the first break are plotted as ordinates against
the composition of the mixture as abscissae, the diagram shown in
Fig. 84 is obtained.

The point A represents the freezing-point of pure silver, B that
of pure copper, the curve AO represents the effect of the gradual
addition of copper upon the freezing-point of silver, and BO the
effect of silver on the freezing-point of copper. The intersection
of the two curves at 0 corresponds to an alloy containing 40 atomic
per cent of copper. This lowest melting mixture is known as
the eutectic (c? = well, and -n/icciv = melt) mixture. At 0 the
system is non-variant, silver, copper, solution and vapor being in
equilibrium. The solid which separates at 0, having a more
uniform texture than that of all other mixtures of the two com-

* Phil. Trans., 189, 25 (1897).

354

THEORETICAL CHEMISTRY

ponents, is known as the eutectic alloy. When the composition
of a mixture of two metals corresponds to that of the eutectic
alloy, the two rnetals crystallize simultaneously in minute separate

40 at. per cent
Concentration
Fig. 84.

crystals. When examined under the microscope the solid eutectic
alloy will be seen to be a conglomerate of very small crystals,
whereas all of the other alloys of the same metals will be found to
contain large crystals of either one or the other component em-
bedded in the conglomerate. While the composition of the eutec-
tic alloy in the above system is found to correspond very closely
to the formula Ag3Cu2, yet the nature of the equilibrium curves
proves it to be nothing more than a mechanical mixture of the
two metals. The meaning of the diagram will be clearer from a
careful consideration of the phenomena accompanying the cooling
of a mixture of the molten metals.

Take for example, a fused mixture relatively rich in silver. As
the temperature falls, a point will ultimately be reached at which

HETEROGENEOUS EQUILIBRIUM

355

pure silver begins to separate, and since the temperature remains
constant during the solidification, a break occurs in the cooling
curve. This first break corresponds to a point on the curve AO.
As silver continues to separate, the composition of the mixtures
changes along AO, until when 0 is reached, the mixture is satu-
rated with respect to copper, and both metals separate as a con-
glomerate having the same composition as the fused mixture.
The separation of the eutectic alloy causes the second break in
the cooling curve, the temperature remaining constant until the en-
tire mass has solidified. It will be noticed that this system is the
exact analogue of the system — anhydrous salt and water; the eutec-
tic point and the cryohydric point representing identical conditions.
(b) Alloys of Gold and Aluminium. This system has been
studied by Roberts- Austen.* The equilibrium curves in the
concentration temperature diagram, Fig. 85, reveal the existence

Au Composition

Fig. 85.

* Phil. Trans. A., 194, 201 (1900).

356 THEORETICAL CHEMISTRY

of definite compounds, AusAU, Au2Al, and AuAl2, corresponding
to the points D, E and H respectively. The discontinuities at B
and G suggest the possibility of two other compounds, viz., Au4Al
and AuAl. The diagram shows that the following substances
will crystallize in succession from the molten alloy, these being
the different solids with which the liquid mixture is saturated in
its successive stages of equilibrium: —

Curve AB, pure gold at A ;

Curve BC, Au4Al, nearly pure at J5;

Curve CD, Au5Al2 or AusAl3, nearly pure at D;

Curve DEF, Au2Al, pure at E\

Curve FG, AuAl, maximum undetermined,

Curve GHI, AuAl2, pure at H]

Curve IJ, Al, pure at J.

The points C, F, and / represent non-variant systems, the melt-
ing-points of the respective eutectic alloys being 527°, 569°, and
647°. This system in many respects resembles the system — ferric
chloride and water.

Three-component Systems. When three components are pres-
ent, the equilibria become much more complicated. Applying
the formula, C — P + 2 = F, we find that it is necessary to
have five phases co-existent for a non-variant system, four for a
univariant, three for a bivariant, and two for a trivariant. The
most satisfactory method of representing equilibria in three-com-
ponent systems is that in which use is made of the triangular
diagram. The three corners of an equilateral triangle are taken
to represent the pure components, and the composition of any
mixture, expressed in atomic percentages, is represented by the
position of the center of mass of the three components within
the triangle.

For example, in the system, — potassium nitrate, sodium nitrate,
and lead nitrate, carefully investigated by Guthrie,* the three
components are placed at the corners of the triangle shown in

* Phil. Mag., 5, 17, 472 (1884).

HETEROGENEOUS EQUILIBRIUM 357

Fig. 86. The melting-point of pure potassium nitrate is 340°
and that of pure sodium nitrate is 305°. The melting-point of

Fig. 86.

pure lead nitrate cannot be determined since the salt decomposes
before its melting-point is reached. The eutectic mixtures of
the three pairs of salts are represented by the points D, E, and F
respectively. In like manner 0 represents the melting-point of
the non- variant system, — potassium nitrate, sodium nitrate, lead
nitrate, fused mixture of the three salts, and vapor. In order
to represent temperature, use is frequently made of a triangular
prism in which the altitude is taken as the temperature axis, the
resulting surface within the prism representing the variation of
the equilibrium with temperature.*

PROBLEMS.

1. The vapor pressure of solid NH^HS at 25°. 1 is 50.1 cm. Assuming
that the vapor is practically completely dissociated into NH3 arid H2S,
calculate the total pressure at equilibrium when solid NH4HS is allowed

* For a complete treatment of three-component systems as well as for a
clear presentation of the phase rule, the student should consult "The Phase
Rule and Its Applications," by Alexander Findlay.

358 THEORETICAL CHEMISTRY

to dissociate at 25°. 1 in a vessel containing ammonia at a pressure of
32 cm. Am. 59.5 cm.

2. In the partition of acetic acid between CCU and water, the con-
centration of the acetic acid in the CCU layer was c gram-molecules per
liter and in the corresponding water layer w gram-molecules per liter.

c 0.292 0363 0.725 1.07 1.41

w 4.87 5.42 7.98 9.69 10.7

Acetic acid has its normal molecular weight in aqueous solutions. From
these figures show that, at these concentrations, the acetic acid in the
carbon tetrachloride solution exists as double molecules.

3. Acetic acid distributes itself between water and benzene in such a
manner that in a definite volume of water there are 0.245 and 0.314 gram
of the acid, while in an equal volume of benzene there are 0.043 and
0.071 gram. What is the molecular weight of acetic acid in benzene,
assuming it to be normal in water? Ans. 121.3.

4. The salt Na2HPOi.l2 H20 has a vapor pressure of 15° of 8.84 mm.,
and at 17°.3 of 10.53 mm. Calculate the heat of vaporization, i.e., the
thermal change during the loss of 1 mol of water of crystallization by
evaporation. Ans. —12,651 cal.

5. The solubility of boric acid in water is 38.45 grams per liter at 13°,
and 49.09 grams per liter at 20°. Calculate the heat of solution of boric
acid per mol. Ans. —5822 cal.

6. Plot the pressure-temperature diagram for calcium carbonate from
the table given on p. 280, and apply the phase rule.

7. Is it possible to decide by the phase rule whether the eutectic alloy
is a mixture or a compound?

CHAPTER XVII.
CHEMICAL KINETICS.

Velocity of Reaction. In the two preceding chapters we have
considered the equilibrium which is established when the speeds
of the direct and reverse reactions have become equal. We now
proceed to consider the velocity of individual reactions. By far
the greater number of the reactions between inorganic substances
proceed with such rapidity that it is impossible to measure their
velocities. Thus, when an acid is neutralized by a base, the indi-
cator changes color almost instantly. There are a few well-
known reactions which are exceptions to this rule; among these
may be mentioned the oxidation of sulphur dioxide and the de-
composition of hydrogen peroxide. Both of these reactions are
well adapted to kinetic experiments. In organic chemistry, on
the other hand, slow reactions are the rule rather than the excep-
tion. Thus the reaction between an alcohol and an acid forming
an ester and water, proceeds very slowly under ordinary condi-
tions and the progress of the reaction may be easily followed.
By means of the law of mass action it is possible to derive equations
expressing the velocity of a reaction at any moment in terms of
the concentrations of the reacting substances present at that time.

Let the equation

represent a reversible reaction and let a, &, c, and d be the respec-
tive initial concentrations of the reacting substances Ai, A 2, Ai,
and A2'. The velocity of the direct reaction will then be

^j-jb(a-*) (6-x), (1)

where k is the velocity constant, and dx is the infinitely small
increase in the amount of x during the infinitely small interval

360 THEORETICAL CHEMISTRY

of time dt. Similarly the velocity of the reverse reaction will
be

^ = ki(c+x) (d + x). (2)

It is evident that the substances on the right-hand side of the equa-
tion will exert an ever-increasing influence upon the velocity of
the direct reaction, which must accordingly decrease. When,
however, the velocities of the direct and reverse reactions become
equal, equilibrium will be established, and the ratio of the amounts
of the reacting substances on the two sides of the equation will
remain constant. The total velocity due to these opposing reac-
tions will be

H = § ~ ¥ = k (a ~~ x} (b " x} ~ kl "(c + x} (d+ X} (3)
and at equilibrium, when ~-^- = 0,

k (a - x) (b - x) = fci (c +x) (d + x),
or

(c + x) (d + x) k ^ , .

(a-x)(b~x) k, c* w

This equation has been thoroughly tested in the two preceding
chapters. Thus, in the reaction

C2H6OH + CH3COOH ?± CH3COOC2H6 + H2U,

Kc has been shown to have the value, 2.84, at ordinary temper-
atures. The velocity constants of the direct and reverse reactions
have also been determined, the values being, k = 0.000238 and
fci = 0.000815. When these values are substituted in the equa-

tion, T- = K c, we obtain Kc = 2.92, a value which agrees well
KI

with that found by direct experiment. The application of equa-
tion (3) is much simplified by the fact that most reactions proceed
nearly to completion in one direction, so that the term k\ (c + x)
(d + x) will be so small that it may be neglected. We then have

»), (5)

CHEMICAL KINETICS

361

an equation expressing the velocity of the direct reaction in terms
of the concentrations of the reacting substances.

Unimolecular Reactions. The simplest type of chemical
reaction is that in which only one substance undergoes change
and in which the velocity of the reverse reaction is negligible. The
decomposition of hydrogen peroxide is an example of such a reac-
tion. In the presence of a catalyst, such as certain unorganized
ferments or colloidal platinum, hydrogen peroxide decomposes
as represented by the equation,

This reaction is usually allowed to take place in dilute aqueous
solution so that there is no appreciable alteration in the amount
of solvent throughout the entire course of the reaction. Further-
more, the activity of the catalyst remains constant so that the
course of the reaction is wholly determined by the concentration
of the hydrogen peroxide. A very satisfactory catalyst is catalase,
an enzyme derived from blood. The concentration of hydrogen
peroxide present at any time during the reaction can be deter-
mined very simply by removing a definite portion of the reaction
mixture, adding an excess of sulphuric acid to destroy the activity
of the hsemase, and then titrating with a standard solution of
potassium permanganate.
The following table gives the results of such an experiment: —

t (minutes).

a — x,
cc. KMn04.

z«
cc. KMnO<.

46 1

5
10
20
30
50

37.1
29.8
19 6
12 3
5.0

9 0
16 3
26.5
33.8
41.1

0 0435
0.0438
0.0429
0 0440
0 0444

Mean 0.0437

The second column of the table gives the number of cubic
centimeters of the potassium permanganate solution required to
oxidize 25 cc. of the reaction mixture when the time intervals

362 THEORETICAL CHEMISTRY

recorded in the first column have elapsed after the introduction
of the catalyst. Since the numbers in the second column repre-
sent the actual concentration of hydrogen peroxide present at
the end of the successive intervals of time, it is evident that the
difference between these numbers and 46.1 cc. — the initial
concentration of hydrogen peroxide — will give the amounts of
peroxide decomposed in those intervals. These numbers are
recorded in the third column of the table. It will be seen that
as the concentration of the hydrogen peroxide decreases the rate
of the reaction diminishes. Thus, in the first interval of 10 min-
utes, an amount of hydrogen peroxide corresponding to 46.1 —
29.8 = 16.3 cc. of potassium permanganate is decomposed, while
in the second interval of 10 minutes, the amount of hydrogen
peroxide decomposed is equivalent to 29.8 — 19.6 = 10.2 cc. of
potassium permanganate. Since only a single substance is under-
going change, equation (5) simplifies to the following form: —

dx j ( ,
#• = *(«-*>•

It is impossible to apply the equation in this form, since in order
to obtain accurate titrations, dt must be taken fairly large and
during this interval of time a—x would have diminished. Approx-
imate values of k may be obtained by taking the average value
of a — x during the interval dt within which an amount dx of hydro-
gen peroxide is being decomposed. For example, let us take the
interval between 5 and 10 minutes; dx = 16.3 — 9.0 = 7.3 cc.,
dt = 5 min., and the average value of a — x is

37.1 + 29.8

Substituting in the equation

we have
and

33.45 cc.

dx j , ^
— *= fc (a -a),

- k X 33.45,

k - 0.0436.

CHEMICAL KINETICS 363

Similarly taking the next interval between 10 and 20 minutes;
dx = 26.5 — 16.3 = 10.2 cc., dt = 10 minutes, and the average

29 g 4- 19 6
value of a — x is — : — ^ "~ ~ 24.7 cc. Substituting in the

equation as before, we obtain

and

k = 0.0413.

As will be seen these two values of k are not in good agreement,
although the first value of k agrees closely with the mean value
of k given in the fourth column of the table.
In order to apply the equation

dx_ ,
dt-b(P> x)

it must be integrated.*

The integration of this equation may be performed as follows: —

^ = k (a - x),

therefore

dx , ,

= k dt.

& — x

Integrating, we have

/j n>

- / k dt = constant = C.
a — x J

therefore

- log, (a - x) - kt = C.

In order to determine C, the constant of integration, we make
use of the experimental fact that when t = 0, x = 0. Substitut-
ing these values, we have

— log«a = C.
Consequently

log, a — log, (a — x) = kt,
or

1 , a ,
7 log, — ^j- = k.

* The student who is unfamiliar with the Calculus must take the result
of this calculation for granted.

364 THEORETICAL CHEMISTRY

Passing to Briggsian logarithms, we obtain

-log— — = 0.4343 k.
t &a — x

By substituting in this equation the corresponding values of a,
a — x, and t from the preceding table, the values of k given in the
fourth column of the table are obtained.
The equation

dx j , ^.
_ = fc(«-*),

may also be thrown into an exponential form, as follows: —
Since

1 , _ a _ ,
710S<^~T ~x - *>

we may write,

7 . a — x

— Kt = lOge

)

a — x = ae~~kt,
and

In this equation k may be regarded as the fraction of the total
amount of substance decomposing in the unit of time, provided
this unit is so small that the quantity at the end of the time unit
is only slightly different from that at the beginning. The time
required for one-half of the substance to change, is known as the
period of half-change, T, and may be calculated from k by means
of the equation

log 2 = 0.4343 fcT,
therefore

0.6943 J,
k

~ - 1.443 T.

Reactions in which only one mol of a single substance undergoes
change are known as unimolecular reactions, or reactions of the

CHEMICAL KINETICS 365

first order. In a unimolecular reaction, the velocity constant k
is independent of the units in which concentration is expressed.
If, in the integrated equation .

kt = log« ,

a — x

t becomes infinite, then x = a. In other words, for finite values
of ty x must always remain less than a and the reaction will never
proceed to completion.

Another unimolecular reaction which has been thoroughly
investigated, is the hydrolysis of cane sugar. When cane sugar
is dissolved in water containing a small amount of free acid it is
slowly transformed into d-glucose and d-fructose. The velocity
of the reaction is very small and is dependent upon the strength
of the acid added. The progress of the reaction may be very easily
followed by means of the polarimeter. Cane sugar itself is dex-
tro-rotatory, while d-fructose rotates the plane of polarization
more strongly to the left than d-glucose rotates it to the right.
Therefore, as the hydrolysis proceeds, the angle of rotation to the
right steadily diminishes until, when the reaction is complete, the
plane of polarization will be found to be rotated to the left. On
this account the hydrolysis of cane sugar is commonly termed
inversion and the molecular mixture of d-fructose and d-glucose
constituting the product of the reaction is called invert sugar.
Let a0 denote the initial angle of rotation, at the time t = 0,
due to a mols of cane sugar, let a0; denote the angle of rotation
when inversion is complete and let a be the angle of rotation at
any time t] then since rotation of the plane of polarization is pro-
portional to the concentration x, the amount of cane sugar in-
verted, will be

Ofo "- Oi

x = a — ; 7-

OiQ + OQ

In the equation

CaHiAi + H20 <=> CcHuOe + C6H,206,

representing the inversion of cane sugar, the velocity of the
reaction will be, according to the law of mass action, proportional
to the molecular concentrations of the cane sugar and the water.

366

THEORETICAL CHEMISTRY

Since the reaction takes place in the presence of such a large excess
of water, its effect may be considered to be constant. The
velocity of the reaction is then proportional to the active mass
of the sugar alone, or in other words the reaction is umimolecular.
In the differential equation expressing the velocity of a unimolec-
ular reaction,

we have

= k (a — x)y

k = -.

and since a and x are measured in terms of angles of rotation of
the plane of polarization, we have

The following table gives the results obtained with a 20 per cent
solution of cane sugar in the presence of 0.5 molar solution of lactic
acid at 25° C.

t (minutes).

34° 5

1,435

31° 1

0 2348

4,315

25°. 0

0.2359

7,070

20°. 16

0 2343

11,360

13° 98

0 2310

14,170

10° 01

0.2301

16,935

7°. 57

0.2316

19,815

5°. 08

0.2991

29,925

- 1°.65

0.2330

Inf.

-10°. 77

Bimolecular Reactions. When two substances react and the
concentration of each changes, the reaction is bimolecular or of
the second order. Let a and b represent the initial molar con-
centrations of the two reacting substances and let x denote the
amount transformed in the interval of time t\ then the velocity
of the reaction will be expressed by the equation

- - k (a - x) (b - x).

CHEMICAL KINETICS 367

The simplest case is that in which the two substances are present
in equivalent amounts. Under these conditions the velocity
equation becomes

This equation may be integrated as follows: — *

7 j* dx
kdt =

therefore

7 ----- -\i>

(a - z)2

A/ I (tt == I / xo >

Jti Jx, (a-z)2

a -a
or

(k ~~ ^i) (a "~ «&i) (# ~~~#2)

If time be reckoned from the beginning of the reaction, then x\ = 0
and t = 0, and we have

k - -7 • —7-^ — r
2 a (a — x)

If the reacting substances are not present in equivalent amounts
then the velocity equation becomes

-- = k (a - x) (b - a).

Assuming that time is measured from the beginning of the reaction,
the integration of this equation may be performed as follows: —

rt rx dx

Jo Jo (CL — x) (b — x)

Decomposing into partial fractions,

&* = — -- 1 i . — I I >

ct — o L«/O o — x t/o & — xj

* The student who is unfamiliar with the Calculus must take the results
of these calculations for granted.

368 THEORETICAL CHEMISTRY

therefore

7 If. a - xf

fc< = ^Llog'^d'

, . 1 , b (a — x)

- _ -^ '

= - _ -

a — b & a (b —
Or passing to Briggsian logarithms,

0.4343 k = 4( l , , log
— *

4( , , — -

t (a — b) *a(b — x)

The value of fc in a bimolecular reaction is not independent of
the units in which the concentration is expressed, as is the case
with a unimolecular reaction. Suppose that a unit I/nth of
that originally selected is used to express concentration, then the
value of k in the equation

z ! x

fv = - -- -, - r )

t a (a — x)
becomes

nx x

t na - n(a — x) t na(a — x)

Thus, the value of k varies inversely as the numbers expressing
the concentrations.

As an illustration of a bimolecular reaction we may take the
hydrolysis of an ester by an alkali. The reaction

CH3COOC2H5 + NaOH ?± CH3COONa + C2HBOH,

has been studied by Warder,* Reicher,f Arrhenius,t Ostwald §
and others. Arrhenius employed in his experiments 0.02 molar
solutions of ester and alkali. These solutions were placed in
separate flasks and warmed to 25° C. in a thermostat maintained
at that temperature; equal volumes were then mixed, and at
frequent intervals a portion of the reaction mixture was removed

* Berichte, 14, 1361 "(1881).

t Lieb. Ann., 228, 257 (1885).

} Zeit. phys. Chem., i, 110 (1887).

§ Jour, prakt. Chem., 35, 112 (1887).

CHEMICAL KINETICS

369

and titrated rapidly with standard acid. The accompanying
table contains some of the results obtained: —

/ (minutes).

a — x

8 04

5 30

o oieo

4 58

0 0156

3 91

0 0164

3 51

0 0160

3 12

0 0162

Mean 0.0160

The numbers in the second column of the table represent the
concentrations of sodium hydroxide and of ethyl acetate, expressed
in terms of the number of cubic centimeters of standard acid
required to neutralize 10 cc. of the reaction mixture. Owing to
the high velocity of the reaction it is difficult to avoid large experi-
mental errors, nevertheless the values of fc given in the third column
of the table will be observed to differ very slightly from the mean
value.

Reicher investigated the same reaction when the reacting sub-
stances were not present in equivalent proportions. In this case,
the progress of the reaction was followed by titrating definite
portions of the reaction mixture from time to time, the excess of
sodium hydroxide being determined by titrating a portion of
the mixture at the expiration of twenty-four hours, when the
ester was completely hydrolyzed. His results are given in the
following table: —

t (minutes).

fl — X

(alkali).

b-x
(ester).

61 95

47 03

4.89
11.36
29.18
Inf

50.59
42.40
29.35
14 92

35.67
27.48
14.43

0 00093
0 00094
0 00092

370 THEORETICAL CHEMISTRY

Reicher also studied the effect of different bases upon the ve-
locity of the reaction. He found for strong bases approximately
equal values of fc, but for weak bases the values were irregular
and smaller than those obtained with the more completely ionized
bases. Arrhenius pointed out that the hydrolyzing power of a
base is proportional to the number of hydroxyl ions which it
yields. Writing the equation for the above hydrolysis in terms of
ions, we have

CH3COOC2H5 + Na' + OH7 *± CH3COO' + Na' + C2H5OH.

It is evident from this equation that all bases furnishing the same
number of hydroxyl ions should give identical values of k. We
may, therefore, modify the fundamental differential equation as
follows: —

~ = k'a (a -x) (b - a;),

where a is the degree of ionization of the base.

Trimolecular Reactions. When equivalent quantities of three
substances react, the reaction is trimolecular or of the third order.
If the initial molar concentrations of the reacting substances are
denoted by a, b, and c, and if x denotes the proportion of each
which is transformed in the interval of time t, the velocity of the
reaction will be represented by the differential equation

•—• = k (a - x) (b - x) (c - re).

If the substances are present in equivalent amounts, the equation
becomes

fa = k (a _ xy

an expression which is much less difficult to integrate.
The integration of this equation may be performed as follows.*

(a - xr

* The student who is unfamiliar with the Calculus must take the results
of these calculations for granted.

CHEMICAL KINETICS 371

therefore,

—

2 >-

hence

i 1 1]

2L(a-z)2 a2J

t 2 a2 (a -a;)2

When the reacting substances are not taken in equivalent amounts,
the integration of the velocity equation may be performed as
follows: —

— - 1c -

therefore

j~
kdt = 7

Decomposing into partial fractions,

(a — x) (b — x) (c — x) a — x b — x c — x'
Multiplying through by (a — x), we obtain

* >*!/„ _\ 1 ** I ^

(6-s)(c-s) " T ^ -'^ft-x^^T^
Let x = a, then

(a - 6) (c - a)

Similarly, multiplying by (b - x) and (c - x), and then placing
x = b, and x = c, we have

~ (a-b)(b-c)'
and

(6-c)(c-a)
Then we obtain by substitution

f _ dx __ _ 1 f dx

Jo (a — x)(b — x) (c — x) (a — 6) (c — a) Jo a — x
__ 1 C* dx ___ 1 /* dx

(a - b) (b - c) Jo b - x (b - c) (c - a) Jo c - x'

372

Therefore,

THEORETICAL CHEMISTRY

(c~a)

I* (a -6) (b-c) (c-a)

In a trimolecular reaction, k is inversely proportional to the square
of the original concentration.

A typical trimolecular reaction is that between ferric and
stannous chlorides. This reaction, represented by the following
equation

2 Fed* + SnCl2 <=* 2 FeCL2 + SnCU,

has been investigated by A. A. Noyes.* Dilute solutions of the
reacting substances were mixed at constant temperature, and
definite portions of the reaction mixture were removed at meas-
ured intervals of time and titrated for ferrous iron. Before
titrating with a standard solution of potassium permanganate it
was necessary to decompose the* stannous chloride present with
mercuric chloride. The following table gives the results obtained
with 0.025 molar solutions of ferric chloride and stannous chloride.

/ (minutes).

a — x

2 5

'0 02149

0 00351

113

0 02112

0 00388

107

0 01837

0 00663

114

0 01554

0 00946

116

0 01394

0 01106

118

0 01313

0 01187

117

0 01060

0 01440

122

0.00784

0.01716

122

Mean 116

Noyes also found that the velocity of the reaction is accelerated
more by an excess of ferric chloride than by an equal excess of
stannous chloride.

* Zeit. phys. Chem., 16, 546 (1895).

CHEMICAL KINETICS 373

Reactions of Higher Orders. Reactions of the fourth, fifth
and eight orders have recently been investigated, but examples
of reactions of orders higher than the third are extremely rare.
This fact is at first sight surprising since the equations of many
chemical reactions involve a large number of molecules, and we
would naturally expect the order of such reactions to be corre-
spondingly high. For example, the reaction represented by the

equation, 2 PH3 + 4 O2 = P2O6 + 3 H20,

involves six molecules of the substances initially present and,
therefore, we should infer it to be a reaction of the sixth order.

Kinetic experiments by van der Stadt have shown it to be a
bimolecular reaction, the velocity of reaction being proportional
to the concentration of the phosphine and the oxygen. On
allowing the gases to mix slowly by diffusion, it was discovered
that the reaction actually takes place in several successive stages,
the first stage being represented by the equation of the bimolec-
ular reaction

PH3 + 02 = HP02 + H2.

The subsequent changes, involving the oxidation of the products
of this reaction, take place with great rapidity. It is highly
probable that the equations which are ordinarily employed to
represent chemical reactions really represent only the initial and
final stages of a series of relatively simple reactions. Larmor *
has shown that when chemical reactions are considered from the
molecular standpoint, the bimolecular reaction is the most prob-
able. He says, " Imagine a substance, say gaseous for simplicity,
formed by the immediate spontaneous combination of three gas-
eous components A, 5, and C. When these gases are mixed, the
chances are very remote of the occurrence of the simultaneous
triple encounter of an A, a 5, and a C, which would be necessary
to the immediate formation of an ABC] whereas if ever formed,
it would be liable to the normal chance of dissociating by collisions;
it would thus be practically non-existent in the statistical sense.
But if an intermediate combination AB could exist, very tran-
siently, though long enough to cover a considerable fraction of the
* Proc. Manchester Phil. Soc., 1908.

374 THEORETICAL CHEMISTRY

mean free path of the molecules, this will readily be formed by
ordinary binary encounters of A and B, and another binary
encounter of AB with C will now form the triple compound ABC
in quantity/'

Determination of the Order of a Reaction. It has been
shown in the foregoing pages that the time required to complete
a certain fraction of a reaction is dependent upon the order of
the reaction in the following manner: —

(1) In a unimolecular reaction the value of k is independent of
the initial concentration;

(2) In a birnolecular reaction the value of k is inversely pro-
portional to the initial concentration;

(3) In a trimolecular reaction the value of k is inversely pro-
portional to the square of the initial concentration.

Hence, in general, in a reaction of the nth order, the value of
k is inversely proportional to the (n — 1) power of the initial con-
centration. If the value of k is determined with definite concen-
trations of the reacting substances, and then with multiples of
those concentrations, the order of the reaction can be determined
according to the above rules by observing the manner in which k
varies with the concentration.

The order of a reaction may also be readily determined by means
of a graphic method. Thus, to determine the order of a reaction
we ascertain by actual trial which one of the following expressions,
in which C denotes concentration, will give a straight line when
plotted against times as abscissae: —

(1) log C — reaction unimolecular;

(2) 1/C — reaction bimolecular;

(3) 1/C2 — reaction trimolecular;

(4) I/O — reaction n + 1 molecular.

Complex Reaction Velocities. Thus far we have considered
the velocity of reactions which are practically complete. There
are numerous cases, however, in which the course of the reaction
is complicated by such disturbing factors as (1) counter reactions,
(2) side reactions, and (3) consecutive reactions. These disturb-
ing causes will now be considered.

CHEMICAL KINETICS 375

(1) Counter Reactions. In the chemical change represented by
the equation

CH3COOH + C2H&OH *=> CH3COOC2H5 + H20,

the speed of the direct reaction steadily diminishes owing to the
ever-increasing effect of the reverse or counter reaction. Ulti-
mately, when two-thirds of the acid and alcohol are decomposed,
the velocities of the two reactions become equal and a condition
of equilibrium results. Starting with 1 mol of acid and 1 mol
of alcohol, and letting x represent the amount of ester formed,
we have

~~j] == ™ \J- 3s) K> X *

When equilibrium is attained,

By observing the change for any time t, we have

39 T
i *i — J/

Having the values of k/k' and k — fc', the velocity constant fc
of the direct reaction can be determined. The value of k so
obtained has been shown by Knoblauch * to vary in those reac-
tions where the concentration of the hydrogen ion changes.

(2) Side Reactions. When the same substances are capable of
reacting in more than one way with the formation of different
products, the several reactions proceeds side by side. Thus,
benzene and chlorine may react in two ways as shown by the
equations,

(1) C6H6 + C12 = C6H6C1 + HC1,
and

(2)

It is generally possible to regulate the conditions under which
the substances react so as to promote one reaction and retard the
other.

* Zeit. phys. Chem., 22, 268 (1897).

376 THEORETICAL CHEMISTRY

(3) Consecutive Reactions. By consecutive reactions we under-
stand those reactions in which the products of a certain initial
chemical change react, either with each other or with the original
substances to form new substances. Attention has already been
called to the fact that many of our common chemical equations
really represent the summation of a number of consecutive reac-
tions. If the system A is transformed into the system C through
an intermediate system J3, then we shall have the two reactions

(1) A-+B
and

(2) B-*C.

If reaction (1) should have a very much greater velocity than
reaction (2), then the measured velocity of the change from A to
C will be practically the same as that of the slower reaction.
This fact has been illustrated by means of the following analogy,
due to James Walker: — * " The time occupied by the transmission
of a telegraphic message depends both on the rate of transmission
along the conducting wire, and on the rate of progress of the
messenger who delivers the telegram; but it is obviously this
last, slower rate that is of really practical importance in determin-
ing the time of transmission." The saponification of ethyl
succinate may be taken as an illustration of consecutive reactions.
This reaction proceeds in two stages as follows : —

COOC2H5
(1) C2H4<; + NaOH^±C2H4 +C2H5OH,

/ /COONa

(2) C2H4/ + NaOH^C2H/ + C2H5OH.

XCOONa N^OONa

In this case the product of the first reaction reacts with one of the
original substances.

Velocity of Heterogeneous Reactions. It has been shown that
when a solid, such as calcium carbonate, is dissolved in an acid,

* Proc. Roy. Soc., Edinburgh, 22 (1898.)

CHEMICAL KINETICS 377

the rate of solution is dependent upon the surface of contact
between the solid and liquid phases, and also upon the strength
of the acid. If the surface is large so that it undergoes relatively
little change during the reaction, it may be considered as constant.
If S represents the area of the surface exposed and x denotes the
amount of solid dissolved in the time t, the velocity of the reaction
will be represented by the differential equation

dx j a , N
-r:=kS(a — x).

Integrating this equation, we have

7 cr 1 i a

M.-.log.—.

This formula has been tested by Boguski * for the reaction
CaC03 + 2 HC1 = CaCl2 + CO2 + H2O,

and is found to give constant values of k. Furthermore, Noyes and
Whitney f have shown that the rate of solution of a solid in a liquid
at any instant, is proportional to the difference between the con-
centration of the saturated solution and the concentration of the
solution at the time of the experiment.

Velocity of Reaction and Temperature. It is a well-estab-
lished fact that the velocity of a chemical reaction is accelerated
by rise of temperature. Thus, the rate of inversion of cane sugar
is increased about five times for a rise in temperature of 30°. It
has been shown as the result of a large number of observations on
a variety of chemical reactions, that in general the velocity of a
reaction is doubled or trebled for an increase in temperature of
10°. It is of interest to note that the rate of development of
various organisms, such as yeast cells, the rate of growth of the
eggs of certain fishes, and the rate of germination of certain
varieties of seeds is either doubled or trebled for a rise in temper-
ature of 10°. Up to the present time no wholly satisfactory form-
ula, connecting the rate of reaction with the temperature, has been
derived, although several purely-empirical expressions have been

* Berichte, 9, 1646 (1876).

t Zeit. phys. Chem., 23, 689 (1897).

378

THEORETICAL CHEMISTRY

suggested. Of these formulas the most widely applicable is that
proposed by van't Hoff and verified by Arrhenius. If fc0 and k\
represent the velocity constants at the respective temperatures
To and Ti, then

where e is the base of the Naperian system of logarithms and A is
a constant. The following table gives the calculated and observed
values of k at various temperatures for the reaction

/NH2

NH4CNO^±OC<

when T = 273 + 25°, k = 0.000227 and A = 11,700.

Tt
Degrees.

K (observed).

Jk (calculated).

273 + 39

0 00141

0 00133

273 + 50 1

0 00520

0.00480

273 + 64 5

0.0228

0.0227

273 + 74 7

0 062

0 0623

273 + 80

0 100

0.105

In this case the agreement between the observed and calculated
values is all that could be desired.

Influence of the Solvent on the Velocity of Reaction. The
velocity of a chemical reaction varies greatly with the nature of
the medium in which it takes place. This subject has been
studied by Menschutkin * who has collected much valuable data,
as the result of a large number of experiments, on the velocity of
the reaction between ethyl iodide and triethylamine, as represented
by the equation

C2H6I + (C2H6)3N = (C2H5)4NL

This reaction was allowed to take place in a large number of
different solvents and the velocity at 100° was measured. A few

* Zeit. phys. Chem,, 6, 41 (1890).

CHEMICAL KINETICS

379

of Menschutkin's results are given in the accompanying table, in
which k denotes the velocity constant: —

Medium.

Hexane

0 00018

Ethyl alcohol

0 0366

Ethyl ether . .

0 000757

Methyl alcohol

0 0516

Benzene

0 00584

Acetone

0 0608

These figures show that the velocity of the reaction is greatly
modified by the nature of the medium in which it takes place, the
velocity in hexane being less than one three-hundreth of that in
acetone. It is of interest to note that there is an approximate
parallelism between the values of fc, and the values of the dielec-
tric constant of the different media.

Catalysis. It is a familiar fact that the velocity of reaction
is frequently greatly accelerated by the presence of a foreign sub-
stance which apparently does not participate in the reaction, and
which remains unchanged when the reaction is complete. For
example, cane sugar is inverted very slowly by pure water alone,
but when a trace of acid is added the reaction is greatly acceler-
ated. A substance which is capable of exerting such an acceler-
ating action is termed a catalyst, and the process is known as
catalysis. In addition to the fact that a relatively-small amount
of a catalyst is capable of effecting the transformation of large
amounts of material, there are two other important character-
istics of catalytic action which should be mentioned: viz., (a) a
catalyst does not initiate a reaction but simply promotes it; and
(6) the equilibrium is not disturbed by the presence of a catalyst,
since the velocities of the direct and reverse reactions are each
altered to the same extent. As the result of a series of experi-
ments, Ostwald concludes that the catalytic effect of acids in
hastening the inversion of cane sugar is directly proportional to
the concentration of the hydrogen ion, and, in general, is inde-
pendent of the nature of the anion. Similarly, the catalytic action
of bases may be attributed to the hydroxyl ion, the effect being
proportional to the concentration of this ion. In fact we may

380 THEORETICAL CHEMISTRY

formulate the following fundamental law of catalysis: — The
degree of catalytic action is directly proportional to the concentration
of the catalytic agent. Almost every chemical reaction can be
accelerated by the addition of an appropriate catalyst. A few
typical reactions which are accelerated catalytically are here
given, together with the catalyst employed : —

Catalyst — hydrogen ion,

CH3COOC2H5 + H2O = CH3COOH + C2H5OH,

Catalyst — hydroxyl ion,

2 CH3-CO-CH3 = CH3CO-CH2C(CH3)2OH,

Catalyst — finely divided platinum,

2 SO2 + 02 = 2 SO3,
2 CH.OH + O2 = 2 H-COH + 2 H2O,
2 H2 + O2 = 2 H2O,

Catalyst — water vapor,

2 CO + O2 = 2 CO2,
NH4C1 = NH3 + HC1,

Catalyst — copper sulphate,

4 HC1 + 02 = 2 H20 + 2 C12,

(Deacon Process)

Catalyst — mercury salts,

2 C10H8 + 9 02 = 2 C6H4/ + 2 H20 + 4 CO2

\COOH

(First step in the synthesis of indigo;

Catalyst — colloidal platinum,

2 H202 = 2 H20 + O2,

Catalyst — enzymes,

C6H1206 = 2 C2H5OH + 2 CO2,

(zymase)

C2H8OH = C3H7COOC2H6 + H2O.

(lipase)

CHEMICAL KINETICS 381

It will be seen that catalysis is of great importance in connection
with many industrial processes as well as in the field of pure
chemistry. The majority of the reactions occurring within
living organisms are accelerated catalytically by unorganized
ferments or enzymes. Thus, before the process of digestion can
proceed, starch must be changed into sugar. This transformation
is accelerated by an enzyme called ptyalin occurring in the saliva,
and by other enzymes found in the pancreatic juice. The digestion
of albumen is hastened by the enzymes, pepsin and trypsin. As
a rule each enzyme acts catalytically on just one reaction, or in
other words the catalytic action of enzymes is specific. Enzymes
are very sensitive to traces of certain toxic substances such as
hydrocyanic acid, iodine, and mercuric chloride.

An interesting series of experiments by Bredig * on the catalytic
action of colloidal metals, established the fact that these sub-
stances resemble the enzymes very closely in their behavior.
Thus, they are " poisoned " by the same substances which inhibit
the activity of the enzymes, and they show the same tendency to
recover when the amount of the poison does not exceed a certain
limiting value. Because of this close similarity, Bredig called the
colloidal metals inorganic ferments.

It sometimes happens that one of the products of a chemical
reaction functions as a catalyst to the reaction. Thus, when
metallic copper is dissolved in nitric acid, the reaction proceeds
slowly at first and then, after a short interval, the speed of the
reaction is greatly augmented. The acceleration is due to the
catalytic action of the nitric oxide evolved. This phenomenon
is known as autocatalysis. In reactions where autocatalysis
occurs,the velocity increases with the time until a certain maximum
value is reached, after which the velocity steadily diminishes. In
ordinary reactions the initial velocity is the greatest.

It sometimes happens that the speed of a reaction is retarded
by the presence of a trace of some foreign substance. Thus,
Bigelow f has shown that the rate of oxidation of sodium sulphite
is retarded by the presence in the solution of only one one-hundred-

* Zeit. phys, Chem., 31, 258 (1899).
t Zeit. Dhvs. Chem.. 26. 493 (1898).

382 THEORETICAL CHEMISTRY

and-sixty-thousandth of a formula weight of mannite per liter.
Such a substance is termed a negative catalyst.

Mechanism of Catalysis. As to the cause of catalytic action
very little is known. In fact it is more reasonable to suppose that
the mechansim of catalysis varies with the nature of the reaction
and the nature of the catalyst, than to conceive all catalytic effects
to be traceable to a common origin. One of the earliest hypotheses
as to the mechanism of catalysis was put forward by Liebig. He
suggested that the catalyst sets up intramolecular vibrations which
assist chemical reaction. The vibration theory was gradually
abandoned as its inadequacy came to be recognized. Of the many
explanations which have been offered to account for catalytic
acceleration, that involving the formation of hypothetical inter-
mediate compounds with the catalyst has been accepted with the
greatest favor. Thus, if a reaction represented by the equation

A + B = AB,

takes place very slowly under ordinary conditions, it is possible
to accelerate its velocity by the addition of an appropriate cat-
alyst (7. According to the theory of intermediate compounds,
the catalyst is supposed to act in the following manner: —

(1) A + C = AC,
(2)

As will be seen, the catalyst is regenerated in the second stage
of the reaction. In 1806 Clement and Desormes suggested that
the action of nitric oxide in promoting the oxidation of sulphur
dioxide in the manufacture of sulphuric acid was purely catalytic.
As is well known, the rate of the reaction represented by the
equation

2 SO2 + O2 = 2 S03,

is very slow. The accelerating action of nitric oxide on the
reaction may be represented in the following manner: —

(1) 2NO + O2 = 2N02,
and

(2) S02 + N02 - S03 + NO.

CHEMICAL KINETICS 383

This explanation, first offered by Clement and Desormes, is still
regarded as the most plausible explanation of the part played by
the oxides of nitrogen in the synthesis of sulphuric acid. It is
apparent that this so-called explanation is far from complete. In
fact, it must be admitted that we have no adequate explanation
of the phenomenon of catalysis. When we are able to answer
the question — "Why does a chemical reaction take place?" —
then we may be able to explain the accelerating and retarding
influences of certain foreign substances on the speed of reactions.
Ostwald likens the action of a catalyst to that of a lubricant on a
machine — it helps to overcome the resistance of the reaction.
If the velocity of a reaction is represented by an equation similar
to that expressing Ohm's law, we have

, ., f , . driving force

velocity of reaction = ~ •

resistance

The driving force is the same thing as the free energy or chemical
affinity of the reacting substances; of the resistance we know
practically nothing. The velocity, according to the above expres-
sion, can be increased in either of two ways, viz., (1) by increas-
ing the driving force, or (2) by diminishing the resistance. It is
inconceivable that a catalyst can exert any effect upon the chem-
ical affinity of the reacting substances, so that we are forced to
conclude that its action must be confined to lessening the
resistance.*

PROBLEMS.

1. When a solution of dibromsuccinic acid is heated, the acid decom-
poses into brom-maleic acid and hydrobromic acid according to the
equation

CHBr-COOH CH-COOH

| =|| + HBr.

CHBr-COOH CBr-COOH

* For an excellent review of the subject of catalysis the student is advised
to consult "Die Lehre von der Reaktionsbeschleunigung durch Fremdstoffe,"
by W. Herz. Ahrens* "Sammlung chemischer and chemisch-technischer
Vortraege."

384

THEORETICAL CHEMISTRY

At 50° the initial titre of a definite volume of the solution was jT0 =
10.095 cc. of standard alkali. After t minutes the titre of the same
volume of solution was Tt cc. of standard alkali.

t 0 214 380

Tt 10.095 10.37 10.57

(a) Calculate the velocity-constant of the reaction.

(b) After what time is one-third of the dibromsuccinic acid decom-
posed? An*, (a) 0.000260; (b) 1559 minutes.

2. From the following data show that the decomposition of H202 in
aqueous solution is a unimolecular reaction: —

Time in minutes 0 10 20

n 22 8 13.8 8.25cc.

n is the number of cubic centimeters of potassium permanganate required
to decompose a definite volume of the hydrogen peroxide solution.

3. In the saponification of ethyl acetate by sodium hydroxide at 10°,
y cc. of 0.043 molar hydrochloric acid were required to neutralize 100 cc.
of the reaction mixture t minutes after the commencement of the reaction.

t 0 4 89 10 37 28 18 infinity

y 61.95 50.59 42.40 2935 14.92

Calculate the velocity-constant when the concentrations are expressed
in mob per liter. Ans. Mean value of k = 2.38.

4. The velocity-constant of formation of hydriodic acid from its ele-
ments is 0.00023; the equilibrium constant at the same temperature is
0.0157. What is the velocity-constant of the reverse reaction?

Ans. 0.0146.

5. Determine the order of the following reaction: —

6 FeCl2 + KC103 + 6 HC1 = 6 FeCls + KC1 + 3 H20.
When the initial concentration of the reacting substances is 0.1, the
changes in concentration at successive times are as follows :

Time (minutes).

Change in Ccn
centration.

0 0048

0 0122

0 0238

0 0329

110

0 0452

170

0 0525

Ans. Third order.

CHAPTER XVIII.
ELECTRICAL CONDUCTANCE.

Historical Introduction. In a book of this character it is
impossible to give anything like a complete historical sketch of
electrochemistry. Before entering upon an outline of this inter-
esting division of theoretical chemistry, however, it is desirable
to consider very briefly a few of the theories which have played a
prominent part in the development of our modern views concern-
ing electrochemical phenomena. While the early observations of
Beccaria and others pointed to the probability of the existence
of some relation between chemical and electrical phenomena, it
was not until the beginning of the nineteenth century that the
science of electrochemistry had its birth. The epoch-making
discovery by Volta of a means of obtaining electrical energy from
chemical energy, gave the initial impulse to all the brilliant dis-
coveries and investigations upon which the modern science of
electrochemistry is based. The apparatus devised by Volta,
known as the voltaic pile, consisted of disks of zinc and silver
placed alternately over one another, the silver disk of one pair
being separated from the zinc disk of the next by a piece of
blotting paper moistened with brine. Such a pile, if composed
of a sufficient number of pairs of disks, will produce electricity
enough to give a shock, if the top and bottom disks, or wires
connected with them, be touched with the moist fingers. This
discovery placed in the hands of the investigator a source of
electricity by means of which experiments could be performed
which had hitherto been impossible. Shortly after the discovery
of the voltaic pile, Nicholson and Carlisle * effected the decom-
position of water, and Davyf isolated the alkali metals As
a result of these experiments, Davy was led to formulate his

* Nich. Jour., 4, 179 (1800).

t Ibid., 4, 275, 326 (1800); Gilb, Ann., 7, 114 (1801).
385

386 THEORETICAL CHEMISTRY

electrochemical theory. According to this theory, the atoms of
different substances acquire opposite electrical charges by con-
tact, and thus mutually attract each other. If the differences
between the charges are small, the attraction will be insufficient
to cause the atoms to leave their former positions; if it is great,
a rearrangement of the atoms will occur and a chemical com-
pound will be formed. In terms of this theory, electrolysis con-
sists in a neutralization of the charges upon the atoms.

The theory of Davy was soon superseded by that of Berzelius.*
According to the latter theory, every atom is charged with both
kinds of electricity which exist upon the atoms in a polar arrange-
ment, the electrical behavior of the atom being determined by the
kind of electricity which is in excess. Chemical attraction is merely
the electrical attraction of oppositely-charged atoms. Since each
atom is endowed with both positive and negative electrification,
one charge being in excess, it follows that the compound formed
by the union of two or more atoms will be positively or negatively
charged according to whichever charge remains unneutralized after
the atoms have combined. Two compounds, the one charged pos-
itively and the other negatively, may thus in turn combine, a
more complex compound being formed. Shortly after Berzelius
formulated his theory, it became the subject of much discussion
and was severely critized. Thus, it was pointed out that if
chemical combination results from the neutralization of oppo-
sitely-charged atoms, then as soon as the charges have become
equalized, there no longer exists any attractive force and the com-
pound must again decompose. This objection was easily overcome
by assuming that as soon as the union between the atoms is
broken, they again acquire their original charges and, in conse-
quence, recombine. In other words, a chemical compound is to
be regarded as existing in a state of unstable equilibrium. An-
other, and apparently insurmountable, objection to the -theory
resulted from the exceptions presented by acetic acid and some of
its substitution products.

According to the theory of Berzelius, chemical combination is
entirely dependent upon the nature of the electrical charges resid-
* Gilb. Ann., 27, 270 (1807).

ELECTRICAL CONDUCTANCE 387

ing on the atoms. From this statement it follows that the prop-
erties of a chemical compound must be a function of the electrical
charges upon the atoms of its constituents. It was shown that
when the three hydrogen atoms of the methyl group in acetic
acid are successively replaced by chlorine, the chemical properties
of the original substance are not materially altered. According
to Berzelius, the three hydrogen atoms are positively charged
while the three chlorine atoms are negatively charged. That
three negative charges could be substituted for three positive
charges in acetic acid without producing a more marked change
in its properties, could not be satisfactorily accounted for by the
theory. This criticism was for a long time considered as an
insuperable barrier to the acceptance of the theory. Shortly
before the close of the nineteenth century, J. J. Thomson * showed
that this objection has little or no weight. When hydrogen gas
is electrolyzed in a vacuum-tube and the spectra at the two elec-
trodes are compared, Thomson found them to differ widely.
From this he concluded that the molecule of hydrogen gas is in
all probability made up of positively- and negatively-charged parts
or ions. He then extended his experiments to the vapors of cer-
tain organic compounds. In discussing these experiments he
says: — "In many organic compounds, atoms of an electro-
positive element, hydrogen, are replaced by atoms of an elec-
tronegative element, chlorine, without altering the type of the
compound. Thus, for example, we can replace the four hydrogen
atoms in CH4 by chlorine atoms, getting, successively, the com-
pounds CHsCl, CH2C12, CHC13, and CC14. It seemed of interest
to investigate what was the nature of the charge of electricity on
the chlorine atoms in these compounds. The point is of some
historical interest, as the possibility of substituting an electro-
negative element in a compound for an electropositive one was
one of the chief objections against the electrochemical theory of
Berzelius."

"When the vapor of chloroform was placed in the tube, it was
found that both the hydrogen and chlorine lines were bright on
the negative side of the plate, while they were absent from the

* Nature, 52, 451 (1895).

388 THEORETICAL CHEMISTRY

positive side, and that any increase in brightness of the hydrogen
lines was accompanied by an increase in the brightness of those
due to chlorine. The appearance of the hydrogen and chlorine
spectra on the same side of the plate was also observed in methyl-
ene chloride and in ethylene chloride. Even when all the
hydrogen in methane was replaced by chlorine, as in carbon tetra-
chloride, the chlorine spectra still clung to the negative side of
the plate. The same point was tested with silicon tetrachloride
and the chlorine spectrum was brightest on the negative side of
the plate. From these experiments it would appear, that the
chlorine atoms in the chlorine derivatives of methane are charged
with electricity of the same sign as the hydrogen atoms they
displace. "

Electrical Units. In 1827, Dr. G. S. Ohm enunciated his well-
known law of electrical conductance, viz.: — The strength of the
electric current flowing in a conductor is directly proportional to the
difference of potential between the ends of the conductor, a?id inversely
proportional to its resistance. If C represents the strength of the
current, E the difference of potential, and R the resistance, then
Ohm's law may be formulated thus: —

C = — -
C R

The unit of resistance is the ohm, that of difference of potential
or electromotive force, the volt, and that of current, the ampere.
The ohm is defined as the resistance of a column of mercury
106.3 cm. long and 1 sq. mm. in cross section at 0° C.s The
ampere is defined as the current which will cause the deposition
of 0.001118 gram of silver from a solution of silver nitrate in 1
second. The volt may be defined as the electromotive force
necessary to drive a current of 1 ampere through a resistance of
1 ohm. The unit quantity of electricity is the coulomb. This
amount of electricity passes when a current of a strength of one
ampere flows for one second. One gram equivalent of any ion
carries 96,500 coulombs, a quantity of electricity known as the
faraday = F . As has already been pointed out, any form of
energy may be considered as the product of two factors, a capac-
ity factor and an intensity factor.

ELECTRICAL CONDUCTANCE 389

The capacity factor of electrical energy is the coulomb while
the intensity factor is the volt, i.e.,

electrical energy = coulombs X volts.

The unit of electrical energy, therefore, is the volt-ampere-second
commonly called the watt-second. One watt-second is the elec-
trical work done by a current of 1 ampere flowing under an elec-
tromotive force of 1 volt for 1 second, and is equivalent to 1 X 107
C.G.S. units. The thermal equivalent of electrical energy may be
calculated from the relation

electrical energy in absolute units , , . - , ,
— ,&£ — . — = heat equiv. of elect, energy,

mechanical equiv. of heat
or

42^«T = 0-2394 cal. = l watt-second.

Faraday's Laws. When two platinum plates or electrodes, one
connected to the positive and the other to the negative terminal
of a battery, are immersed in a solution of sodium chloride, it
will be found that hydrogen is immediately evolved at the nega-
tive electrode and oxygen at the positive electrode. If the salt
solution is previously colored with a few drops of a solution of
litmus it will be observed that the portion of the solution in the
neighborhood of the positive electrode will turn red, indicating
the formation of an acid, while that in the neighborhood of the
negative electrode will turn blue, showing the formation of a
base. The same changes will take place whether the electrodes
are placed near together or far apart, and furthermore, the evolu-
tion of gas and the change in color at the electrodes commences
as soon as the circuit is closed. The study of these phenomena
led Faraday * to the conclusion, that when an electric current
traverses a solution, there occurs an actual transfer of matter,
one portion travelling with the current and the other portion
moving in the opposite direction. At the suggestion of the philol-
ogist Whewell, Faraday termed these carriers of the current, ions
= to wander). He also called the electrode connected to
* Experimental Researches (1834).

390

THEORETICAL CHEMISTRY

the positive terminal of the battery, the anode (dm = up and
£809 = way), and the electrode connected to the negative terminal
the cathode (/cara = down and 6805 = way). The ions which
move toward the anode he called anions, while those which
migrate toward the cathode he called cations. The whole process he
termed electrolysis. The question of the relationship between the
amount of electrolysis and the quantity of electricity passing
through a solution was investigated by Faraday. As a result of
his experiments he enunicated the following laws which are com-
monly known as the laws of Faraday: —

(1) For the same electrolyte, the amount of electroysis is propor-
tional to the quantity of electricity which passes.

(2) The amounts of substances liberated at the electrodes when the
same quantity of electricity passes through solutions of different
electrolytes, are proportional to their chemical equivalents. The
chemical equivalent of any ion is equal to the atomic weight divided
by its valence. If the same quantity of electricity is passed
through solutions of hydrochloric acid, silver nitrate, cuprous
chloride, cupric chloride, and auric chloride, the relative amounts
of the different cations liberated will be as follows: —

Electrolyte.

Chem. Equiv. of
Cation.

HC1

H' = 1

AgNOs

Ag' « 108

Cu2Cl2

Cu' = 63.4

CuCl«

Cu" =63.4-r2

AuCl3

Aif" = 197-^3

The electrochemical equivalent of an element or group of elements
is the weight in grams which is liberated by the passage of one
coulomb of electricity. The electrochemical equivalents are,
according to Faraday's second law, proportional to the chemical
equivalents. The quantity of electricity necessary to liberate one
chemical equivalent in grams is called a faraday. This is a very
important unit in electrochemical calculations. Since one coulomb
liberates 0.00001036 gram of hydrogen, 1 -5- 0.00001036 = 96,500

ELECTRICAL CONDUCTANCE 391

coulombs of electricity will be required to liberate one gram equiv-
alent of hydrogen. The same quantity of electricity will liberate
35.45 X 0.00001036 = 0.000368 gram of chlorine, and 108 X
0.00001036 = 0.001118 gram of silver. Or, in general, since one
coulomb of electricity liberates 0.00001036 gram of hydrogen, it
will cause the liberation of 0.00001036 w grams of any other ele-
ment whose equivalent weight is w.

The Existence of Free Ions. When an electrolyte is de-
composed by the electric current, the products of decomposition
appear at the electrodes. The fact that the liberation of the prod-
ucts of decomposition is independent of the distance between
the electrodes caused considerable difficulty in the early history
of electrolysis. It was evident that the two products could
hardly be derived from the same molecule, but must come from
two different molecules. Several theories were advanced to
account for the experimental results. Thus, in the electrolysis of
water it was suggested that the two gases, hydrogen and oxygen,
were not derived from the water but that electricity itself pos-
sessed an acid character. Grotthuss * was the first to propose a
rational hypothesis as to the mechanism of electrolysis. He
assumed that when the electrodes in an electrolytic cell are con-
nected with a source of electricity, the molecules of the electrolyte
arrange themselves in straight lines between the electrodes, the
positive poles being directed toward the negative electrode and
the negative poles toward the positive electrode. When elec-
trolysis begins, the cation of the molecule nearest the cathode
is liberated at the cathode and the anion of the molecule nearest
the anode is liberated at the anode. The anion which is left
free near the cathode then combines with the cation of the next
adjoining molecule, the anion thus left uncombined uniting with
the cation of its nearest neighbor, a similar exchange of partners
continuing throughout the entire molecular chain. Under the
directive influence of the two electrodes, the newly-grouped mole-
cules then rotate so that the positive poles all face the negative
electrode and the negative poles all face the positive electrode.
The process is then repeated, another molecule being electrolysed.

* Ann. de Chim. [1], 58, 54 (1806).

392 THEORETICAL CHEMISTRY

This theory of electrolysis appears to have been accepted by
Faraday. Its inherent defect was first pointed out by Grove.*
From his experiments with the oxy-hydrogen cell, which derives
its energy from the union of hydrogen and oxygen, he pointed out
that a decomposition of the molecules of water is not essential
for the evolution of these two gases, but that the molecules must
be already in a state of partial decomposition. This suggestion
was followed up by Clausius. f He argued that if an expenditure
of energy is necessary to decompose the molecules, electrolysis
should be impossible at very low voltages. Experiment showed
that when silver nitrate is electrolyzed between silver electrodes,
decomposition takes place at voltages which are much below the
voltage corresponding to the energy of formation of silver nitrate.
In other words, it requires very little energy to decompose a salt
which is formed with the evolution of a large amount of energy,
a result which is in contradiction to the principle of the conserva-
tion of energy. Clausius was thus forced to conclude "that the
supposition that the constituents of the molecule of an electrolyte
are firmly united and exist in a fixed and orderly arrangement is
wholly erroneous."

As a result of his investigation of the synthesis of ethyl ether
from alcohol and sulphuric acid, Williamson f concluded "that in
an aggregate of the molecules of every compound, a constant inter-
change between the elements contained in them is taking place."
In the same paper he writes, "each atom of hydrogen does not
remain quietly attached all the time to the same atom of chlorine,
but they are continually exchanging places with one another."
This view was accepted by Clausius, although he had no means of
determining the extent to which the electrolyte was broken down
or dissociated into free ions.

In 1887, Arrhenius § developed the views of Clausius by showing
how the degree of dissociation of the molecules of an electrolyte
can be deduced from measurements of the electrical conductance

* Phil. Mag., 27, 348 (1845).
t Pogg. Ann., 101, 338 (1857).
t Lieb. Ann., 77, 37 (1851).
§ Zeit. phys. Chem., i, 631 (1887).

ELECTRICAL CONDUCTANCE

393

of its solutions, as well as from measurements of osmotic pressure
and freezing-ptfint lowering. The important generalization sum-
marizing these conceptions is known as the theory of electrolytic
dissociation, to. which reference has already been made in earlier
chapters (see page 227).

The Migration of the Ions. Since the passage of a current of
electricity through a solution of an electrolyte causes the dis-
charge of equivalent amounts of positive and negative ions at the
electrodes, it might be inferred that the ions all move with the
same speed. That this inference is incorrect, was first shown by
Hittorf * as the result of his observations on the changes in con-
centration of the solution in the neighborhood of the electrodes

00®® ®l® ©

O 0 0 010 O 0 0 0 0j0 0 O O
I n !

0 0 0 0 0 0|0 0 0 0 0 00 0

» III J

00000 010 0000 0[0 0

Fig. 87.

during electrolysis. He showed that different ions migrate with
different speeds, and that the faster moving ions carry a greater
proportion of the current than the slower moving ions. The
effect of unequal ionic velocities on the concentrations of the
solutions around the electrodes is clearly shown by the ac-
companying diagram (Fig. 87) due to Ostwald. The anode and

* Pogg. Ann., 89, 177; 98, 1- 103, 1; 106, 337, 513 (1853-1859).

394 THEORETICAL CHEMISTRY

cathode in an electrolytic cell are represented by the vertical lines
A and C respectively. The cell is divided into three compart-
ments by means of porous diaphragms,* represented by the ver-
tical dotted lines. The cations are represented by dots (*) and
the anions by dashes ('). Before the current passes through the
cell, the concentration of the solution is uniform throughout, the
conditions being represented by I. Now let us imagine that only
the anions move when the current is established. The conditions
when the chain of anions has moved two steps toward the anode
are shown in II. Each ion which has been deprived of a partner
is supposed to be discharged. It will be observed that although
the cations have not migrated toward the cathode, yet an equal
number of positive and negative ions are discharged, and that
while the concentration in the anode compartment has not changed,
the concentration in the cathode compartment has diminished to
one-half its original value.

Let us now suppose that both anions and cations move with the
same speed, and as before, let each chain of ions move two steps
toward their respective electrodes, as indicated in III. It will be
seen that four positive and four negative ions have been dis-
charged, and that the concentration of the electrolyte in the anode
and cathode compartments has diminished to the same extent.
Finally, let us assume that the ratio of the speeds of the cations
to that of the anions is as 3 : 2. When the cations have moved
three steps toward the cathode and the anions have moved two
steps toward the anode, the conditions will be as shown in IV.
It is evident that five positive and five negative ions have been
discharged, and that the concentration in the cathode compart-
ment has diminished by two molecules while the concentration
in the anode compartment has diminished by three molecules.
It will b*e observed that the change in concentration in either of
the electrode compartments is proportional to the speed of the
ion leaving it. Thus, in II, the concentration in the cathode
compartment diminishes while that in the anode compartment
remains unchanged, since only the anion moves. In like manner,
the change in concentration about the electrodes in III corre-
sponds with the fact that both ions migrate «.t the Same rate.

ELECTRICAL CONDUCTANCE 395

In IV the ratio of the change in concentration in the cathode
compartment to that in the anode compartment is as 2 : 3. It
will be apparent from tfyese examples, that the relation between
the speeds of the ions and the corresponding changes in concen-
tration at the electrodes may be expressed by the following pro-
portion: —

Change in concentration at anode __ speed of cation
Change in concentration at cathode speed of anion

If the relative speed of the cations is represented by u, and that
of the anions by v, then the total quantity of electricity trans-
ported will be proportional to u + v: of this total, the fractions

carried by the anion and cation respectively, will be n = — : — ,

and 1 — w = — -p — The values of these ratios, n and 1 — n,
u ~i v

are called the transport numbers of the anion and cation respec-
tively. It is apparent from the diagram, that if the electrolysis
is not carried too far, the concentration of the solution in the inter-
mediate compartment will undergo no change. In order to deter-
mine transport numbers, therefore, it is simply necessary to
remove portions of the solutions in the immediate vicinity of the
two electrodes and determine the concentration of the electrolyte
analytically. The success of the experiment depends upon keep-
ing the concentration of the intermediate compartment unaltered.
Experimental Determination of Transport Numbers. Various
forms of apparatus have been constructed for the determination
of transport numbers, among which one of the most satisfactory
is that devised by Jones and Bassett,* and shown in Fig. 88. It
consists of two vertical tubes of wide bore connected by a U-tube
fitted with a stop-cock. Into each of two electrodes, made of
a suitable metal, is riveted a short piece of stout platinum wire,
which is then sealed into heavy-walled glass tubes. The exposed
end of the platinum wire on the under side of each electrode is
covered with a drop of fusion glass. The tubes carrying the
electrodes are fitted into holes bored through the ground glass
* Am. Chem. Jour., 32, 409 (1904).

396

THEORETICAL CHEMISTRY

stoppers which close the right and left arms of the apparatus.
Two small graduated tubes are sealed to the two vertical tubes
just below the stoppers. These tubes allow for any slight dis-
placement of the solution due to expansion or the formation of

Fig. 88.

gas, and at the same time make it possible to level the apparatus
accurately. When electrolysis has proceeded far enough, the
circuit is broken and the stop-cock closed, thus preventing the
mixing of the solutions in the anode and cathode compartments.
The solutions in the two halves of the apparatus are then rinsed out
into separate beakers and the concentration of each is determined
analytically. Knowing the initial concentration of the solution
and the final concentrations at the two electrodes, together with
the total quantity of electricity which has passed through the
apparatus during the experiment, we have all of the data neces-
sary for the calculation of the transport numbers of the two ions.

ELECTRICAL CONDUCTANCE 397

The following example will serve to make the method of calcu-
lation clear: — In an experiment to determine the transport
numbers of the ions of silver nitrate, a solution containing 0.00739
gram of that salt per gram of water was prepared. The solution
was introduced into the migration apparatus and, after inserting
silver electrodes, a small current was passed through the appa-
ratus for two hours. A silver coulometer was included in the cir-
cuit, and 0.0780 gram of silver was deposited by the current.

This mass of silver is equivalent to 0.000723 gram-equivalent.
After the circuit was broken, the anode solution was rinsed out and
its concentration determined analytically. It was found to con-
tain 0.2361 gram of silver nitrate to 23.14 grains of water. This
amount of solution contained originally 23.14 X 0.00739 =
0.1710 gram of silver nitrate. Thus, the amount of silver nitrate
in the anode compartment had increased by 0.2361 — 0.1710 =
0.0651 gram of silver nitrate, or 0.000383 gram-equivalent of
silver. Obviously the increase in the concentration of the nitrate
ion must have been the same. The amount of silver dissolved
from the anode must have been equal to that deposited in the
coulometer, or since 0.000723 gram-equivalent of silver was
deposited and the actual increase found was 0.000383 gram-
equivalent, the difference, 0.000723 - 0.000383 = 0.000340 gram-
equivalent, is the amount of silver which migrated away from the
anode. At the same time 0.000383 gram-equivalent of nitrate
ions migrated into the anode compartment. The ratio of the
speed of migration of the silver ions to that of the nitrate ions is
as 0.000340 : 0.000383. - Since 0.000723 gram-equivalent of silver
ions measures the total quantity of electricity transported, the
transport numbers of the two ions will be as follows: —

Transport number of Ag- = 1 - n = Q = 0.470,

Transport number of NO8'= n = £S§| - 0.530.

These numbers can be checked by a similar calculation based on
the change in concentration in the cathode compartment.

398

THEORETICAL CHEMISTRY

The following table gives the transport numbers of the anions
of various electrolytes at different dilutions, V being the number
of liters of solution containing one gram-equivalent of solute.
The transport numbers of the corresponding cations can be found
by subtracting the transport numbers of the anions from unity.

TRANSPORT NUMBERS OF ANIONS.

•y=

100

0.5

KC1 I
KBr

0.506

0 507

0 508

0 509

0.513

0.514

0 515

KI [
NH4C1 J
NaCl

0 614

0 617

0.620

0 626

0 637

KNO,

0 497

0.496

0 492

0.487

0.479

AgNOa

0 528

0.528

0 527

0.519

0 501

0.476

KOH . .

0.735

0 736

HC1

0.172

0 172

0.172

O.i73

0 176

iBaCU ....

0.640

0.657

K2CO3 . .

0 435

0 434

0 413

CuSO4.

0 620

0 626

0.632

6.643

0 668

0 696

0.720

H2SO4

0 182

0 174

It is apparent from the table that the transport numbers are
not entirely independent of the concentration. They also vary
slightly with the temperature and approach the limiting value,
0.5, at high temperatures.

Specific, Molar and Equivalent Conductance. As is well
known, the resistance of a metallic conductor is directly propor-
tional to its length and inversely proportional to its area of cross-
section. Similarly, the resistance of an electrolyte is proportional
to the length and inversely proportional to the cross-section of
the column of solution between the two electrodes. The specific
resistance of an electrolyte may be defined as the resistance in
ohms of a column of solution one centimeter long and one square
centimeter in cross-section. Specific conductance is the reciprocal
of specific resistance. Since the conductance of a solution is
almost wholly dependent upon the amount of solute present, it
is more convenient to express conductance in terms of the molar
or equivalent concentration. The molar conductance n, is the

ELECTRICAL CONDUCTANCE

399

conductance in reciprocal ohms, of a solution containing one mol
of solute when placed between electrodes which are exactly one
centimeter apart. The equivalent conductance A is the conduc-
tance in reciprocal ohms of a solution containing one gram-
equivalent of solute when placed betweep electrodes which are
one centimeter apart. If K denotes the specific conductance of a
solution and Fm, the volume in cubic centimeters which contains
one mol of solute, then

and in like manner

A =

where Ve is the volume of solution in cubic centimeters which
contains one gram-equivalent of solute. The following table
gives the specific and molar conductance of solutions of sodium
chloride at 18° C.: —

Concentration.

Dilution.

Sp. Cond.

Molar Cond.

1,000

0.0744

74 4

0 1

10,000

0.00925

92 5

0.01

100,000

0 001028

102 8

0 001

1,000,000

0.0001078

107.8

0 0001

10,000,000

0.00001097

109 7

It will be observed that the molar conductance increases with the
dilution up to a certain point beyond which it remains nearly
constant. That the molar conductance should change but little
will become apparent from the following considerations: —
Imagine a rectangular cell of indefinite height and having a cross-
sectional area of one square centimeter, and further assume that
two opposite walls can function as electrodes. Let 1000 cc. of a
solution containing one mol of solute be introduced into the cell,
and let its conductance be determined. Now let the solution be
diluted to 2000 cc. and the conductance of the diluted solution be
measured. While the specific conductance of the diluted solution
is reduced to one-half of its original value, yet since the electrode

400

THEORETICAL CHEMISTRY

surface in contact with the solution is doubled, owing to the fact
that the solution stands at twice the original height in the
cell, the total conductance due to one mol of solute remains un-
changed. This, of course, is only the case with completely ionized
solutes.

Determination of Electrical Conductance. The determination
of the electrical conductance of a solution resolves itself into
the determination of its resistance by a simple modification of
the familar Wheatstone-bridge method. The arrangement of the
apparatus for this method devised by Kohlrausch * is represented
diagrammatically in Fig. 89, where ab is the bridge wire, B is a

Fig. 89.

resistance box, and C is a cell containing the solution whose
resistance is to be measured. The points d and c are connected
to a small induction coil / which gives an alternating current.
This is necessary in order to prevent polarization which would
occur if a direct current were used. The use of the alternating
current necessitates the substitution of a telephone, T, for the
galvanometer usually employed in measuring resistance. The
positions of the induction coil and telephone are sometimes inter-
changed, but the arrangement shown in the diagram is to be pre-
ferred, since it insures a high electromotive force where the sliding

* Wied. Ann., 6, 145 (1879); n, 653 (1880); 26, 161 (1885).

ELECTRICAL CONDUCTANCE

401

contact c touches the wire, this being the most uncertain connec-
tion in the entire arrangement. A small accumulator A, serves
to operate the induction coil. In making a measurement, the
coil is connected with the accumulator and the vibrator adjusted
so that a high mosquito-like tone is emitted; then the sliding
contact c is moved along the wire ab until the sound in the tele-
phone reaches a minimum, the position of the point of contact
with the bridge-wire being read on the millimeter scale placed
below. According to the principle of the Wheatstone bridge, it
follows that

C^fa

B~ ac'

Since the resistance B a*id the lengths be and ac are known, the
resistance C can be calculated. Various types of conductance
cells are in use, depending upon whether
the solution has a high or a low resistance.
The form shown in Fig. 90 is widely used.
The two electrodes are made of platinum
foil, connection with the mercury in the two
glass tubes it being established by means of
two pieces of stout platinum wire sealed
through the ends of these tubes. The tubes it
are fastened into a tight-fitting vulcanite
cover so that the electrodes may be re-
moved, rinsed and dried without altering
their relative positions. Before the cell is
used, the electrodes must be coated electro-
lytically with platinum black. It is not
necessary to know the area of the elec-
trodes or the distance between them, since
it is possible to determine a factor, termed
the resistance capacity, by means of which
the results obtained with the cell can be ^ig. 90.

transformed into reciprocal ohms. To this
end the specific conductances of a number of standard solu-
tions have been carefully determined by Kohlrausch; thus, for

402 THEORETICAL CHEMISTRY

a 0.02 molar solution of potassium chloride he found th* Allowing
values: —

K18o = 0.002397 and «* = 0.002768,
or

Al8o = 119.85 and A2Bo = 138.4.

Let the resistance of the cell when filled with 0.02 molar potassium
chloride be C, then according to the principle of the Wheatstone
bridge we have

C = B • — ,
ac

or denoting the conductance of the solution by L, we obtain

T 1 ac

JU = 77 =

C B • be

Since the specific conductance K must be proportional to the
observed conductance, we have

~-^B^bc'

where K is the resistance capacity of the cell. If the measure-
ment is made at 18° C., then we have

= 0.002397 B • be
ac

Having determined the resistance capacity of the cell we may
then proceed to determine the conductance of any solution. For
example, suppose that when the resistance in the box is B', the
point of balance on the bridge-wire is at c', then the specific con-
ductance of the solution will be

* B'bc'

If k' is multiplied by the volume of the solution, we obtain the
equivalent conductance, or

A =

Relative Conductances of Different Substances. The study
of the electrical conductance of various solutes in aqueous solu-
tion, reveals the fact that electrolytes differ greatly in their con-
ducting power. They may be roughly divided into two classes: —

ELECTRICAL CONDUCTANCE

403

those with high conducting power, such as strong acids, strong
bases, and salts; and those with low conducting power, such as
ammonia and most of the organic acids and bases. Further-
more, the equivalent or molar conductance increases with the dilu-
tion until a dilution of about 10,000 liters is reached, beyond
which it remains constant. The following table gives the equiv-
alent conductances of three typical electrolytes, V representing
the volume of the solution in liters, and A the equivalent con-
ductance: —

HYDROCHLORIC ACID.

A (18°)

0 333

201 Q

1 0

278 0

10 0

324 4

100 0

341 6

1000 0

345 5

SODIUM HYDROXIDE.

A (18°).

0 333

100 7

1 0

149 0

10 0

170 0

100 0

187 0

500.0

186.0

POTASSIUM CHLORIDE.

A 08°).

0 333

82 7

1 0

91.9

10 0

104.7

100 0

114.7

1,000 0

119.3

10,0000

120.9

404 THEORETICAL CHEMISTRY

The curves shown in Fig. 91 are plotted from the data of the
foregoing table, and bring out very clearly the differences in con-
ducting power possessed by the three electrolytes.

In general the conductance of pure liquids is small. Thus, the
specific conductance of pure water at 18° is approximately 1 X 10~6

Dilution, V

Fig. 91.

reciprocal ohms and, as Walden * has shown, the specific conduct-
ance of a number of other solvents is of the same order as that
for water. Mixtures of two liquids, each of which is practically
non-conducting, may have a conductance differing but little from
that of the two components; or the mixture may have a very
high conductance. For example, the conductance of a mixture

* Zeit. phys. Chem., 46, 103 (1903).

ELECTRICAL CONDUCTANCE 405

of water and ethyl alcohol is of the same order of magnitude as
that of the two components, while on the other hand, a mixture
of water and sulphuric acid, each of which in the pure state is
practically a non-conductor, has great conducting power. The
variation of the specific conductance of mixtures of water and
sulphuric acid is represented in Fig. 92, the concentrations of sul-
phuric acid being plotted on the axis of abscissae and the specific
conductances on the axis of ordinates. It appears that as the

10 20 30 40 60 60 70 80 90 100 110
Per Cent Sulphuric Acdcl

Fig. 92.

Concentration of the sulphuric acid increases, the specific conduct-
ance of the mixture increases until 30 per cent of acid is present,
Beyond which point it gradually diminishes. When pure sul-
Dhuric acid is present the value of the specific conductance is
practically zero. On dissolving sulphur trioxide in the pure acid,
the specific conductance increases slightly to a maximum and then
falls rapidly to zero. There is a minimum in the curve corre-
sponding to about 85 per cent of acid, a concentration which

406

THEORETICAL CHEMISTRY

corresponds almost exactly with the hydrate H2S04H20. Why
some liquid mixtures should have marked conducting power and
others hardly any, it is difficult to explain. Many fused salts,
such as silver nitrate and lithium chloride, are excellent conductors
and are thus exceptions to the general rule, that pure substances
belonging to the second class of conductors possess little conduct-
ing power.

The Law of Kohlrausch. The electrical conductance of solu-
tions was systematically investigated by Kohlrausch who showed
that the limiting value of the equivalent conductance, which may
be represented by A*, is different for different electrolytes and
may be considered as the sum of two independent factors, one of
which refers to the cation and the other to the anion. This experi-
mental result is commonly known as the law of Kohlrausch.

The limiting value of the equivalent conductance is reached
when the molecules are completely broken down into ions, and
under these conditions the whole of the electrolyte participates
in conducting the current. The accompanying table, giving the
equivalent conductances at infinite dilution of several binary
electrolytes, illustrates the truth of the law of Kohlrausch.

EQUIVALENT CONDUCTANCES AT INFINITE DILUTION.

NH4

Cl .

1231

103

122

353

NO8 . .
OH

ll&i

228

.98'
201

350

109

C108

115

103

C2H302

The differences between two corresponding sets of numbers in
the same vertical column- and of any two corresponding sets of
numbers in the same horizontal, row. will be found to be nearly
equal. This could only occur when the limiting conductance is
the sum of two entirely independent quantities. Each ion
invariably carries the same charge of electricity and moves with

ELECTRICAL CONDUCTANCE 407

its own velocity quite independent of the nature of its compan-
ion ion. Therefore^ at infinite dilution, we have

in which lc and la are the equivalent conductances of the ions of
the electrolyte at infinite dilution. From this it follows that

ypu k - A
n^lc + la~ A.'
and

la =
and ~ —

lc = (1 - n) Aoo.

Thus, the equivalent conductance of silver nitrate at infinite dilu-
tion at 18° is 115.5, while n = 0.518 and 1 - n = 0.482; there-
fore

Z« = 0.518 X 115. 5 = 59.8,
and

lc = 0.482 X 115.5 = 55.7;

or one gram-equivalent of N(V ions possesses a conductance of
59.8 when placed between electrodes one centimeter apart and
large enough to contain between them the entire volume of solu-
tion in which the NO3' ions exist; and one gram^equivalent of
Ag" ions under the same conditions have a conductance equal
to 55.7.

The values of the ionic conductances at infinite dilution remain
constant in all solutions in the same solvent at the same temper-
ature, so that it is possible to calculate the equivalent conductance
for any substance at infinite dilution.

In the subjoined table are given the ionic conductances of
various ions at 18° and infinite dilution, together with their temper-
ature coefficients.

408 THEORETICAL CHEMISTRY

IONIC CONDUCTANCES AT, INFINITE DILUTION.

Ion.

Temp. Coeff .

Li*

33 44

0 0265

Na*

43 55

0 0244

64 67

0 0217

Rb*

67 6

0 0214

Cs'

68 2

0 0212

NH4*

64 4

0 0222

Tl*

- 66 0

0 0215

Ag* .*...

54.02

0 0229

46 64

0.0238

Cl'

65.44

0 0216

Br' s.

67.63

0 0215

I'.

66.40

0 0213

SCN'

56 63

0 0211

C1O8'

55 03

0 0215

IO3'

33 87

0 0234

NO8'

61 78

0.0205

318 0

OH'

174.0

| Zn"

45 6

0 0251

| Mg"

46.0

0 0256

|Ba"

56 3

0 0238

J Pb"

61.5

0 0243

JSO4"

68.7

0 0227

\ CO3"

70 0

0 0270

In the case of weak electrolytes the value of A* cannot be
determined directly from conductance measurements, since before
the limiting value is reached, the solution has become so dilute as
to render accurate measurements of the specific conductance
impossible. The law of Kohlrausch enables us to get around
this difficulty. Thus, the value of Aoo for acetic acid must be
equal to the sum of the conductances of the H* and CHsCOO'
ions. The conductance of the H* ion at 18° is, according to
the preceding table, 318. The value of the conductance of the
CHsCOO' ion can be determined from the conductance of sodium
acetate at infinite dilution, Aoo for this salt being 78.1 at 18°.
Since the ionic conductance of the Na* ion is 43.55 at 18°, it
follows that the conductance of the CH3COO' ion must be 78.1 —
43.55 = 34.55. Therefore, for acetic acid we have

Aoo - lc + L - 318 + 34.55 = 352.55 at 18°.

ELECTRICAL CONDUCTANCE

409

Bredig * has shown that the ionic conductance of elementary
ions is a periodic function of the atomic weight. When the ionic

80-
60-
40
80-

80 120

Atomic Weight

Fig. 93.

100

800

conductances are plotted as ordinates against the atomic weights
as abscissae, the curve shown in Fig. 93 is obtained. A glance at
the curve shows the periodic nature of the relation.

Absolute Velocity of the Ions. Thus far we have considered
only the relative velocities of the ions and their conductances;
we now proceed to the consideration of their absolute velocities
in centimeters per second.

Let a current of electricity pass through a centimeter cube of a
solution of a binary electrolyte. If the solution contains m mols
of solute per liter, then w/1000 will be the number of mols in the
centimeter cube. The charge on either the cation or the anion

m : , where F = 96,540 coulombs. If C represents the total

»F1000»
current, we have

, m
1000

V),

Zeit. phys. Chem., 13, 242 (1894).

410 THEORETICAL CHEMISTRY

since the current is the charge which passes through one face of
the cube in one second. In a centimeter cube, the current is
equal to the product of the specific conductance and the difference
of potential E, the latter being numerically equal to the potential
gradient, the distance between the electrodes being one centi-
meter. Hence, we have

1000 KE = Fm (U + V).

If E is expressed in volts and K in reciprocal ohms, U and V will be
expressed in centimeters per second, for on passing to absolute
electromagnetic units, we have

1000 (K X 10-9) (E X 108) ,_7 . .„

- (F3ooH - "- m(u + y>>

where U and V are the ionic velocities for unit potential gradient —
1 volt per centimeter.
From this it follows that

The equivalent conductance of a 0.0001 molar solution of potas-
sium chloride at 18° is 128.9; the total velocity of the two ions
is then,

0.001345 cm. per sec.

This total velocity is made up of the two individual ionic velocities.
The transport numbers of the two ions, K* and Cl;, are respectively
0.493 and 0.507. Hence the absolute velocities of the ions, ex-
pressed in centimeters per second, in a 0.0001 molar solution of
potassium chloride at 18° are as follows: —

U = 0.001345 X 0.493 = 0.00066 cm. per sec.,
and

V = 0.001345 X 0.507 = 0.00068 cm. per sec.

ELECTRICAL CONDUCTANCE

411

The absolute velocities of some of the more common ions at 18C
are given in the following table: —

ABSOLUTE IONIC VELOCITIES.

Ion.

Velocity.

Ion.

Velocity.

cm per sec.
0 00066

cm. per sec.
0.00320

NH4*

0 00066

0 00069

Na*

0 00045

NO3'

0 00064

Li*

0 00036

CUV

0 00057

Ag-
Cr2O7"

0 00057
0 000473

OH'
Cu"

0.00181
0 00031

The velocities of certain ions have been determined directly.
Thus, the velocity of the hydrogen ion was measured by Lodge *
in the following manner: — The tube 5, Fig. 94, 40 cm. long and
8 cm. in diameter, was graduated and bent at right angles at the

Fig. 94.

ends. This was filled with an aqueous solution of sodium chloride
irf gelatine, colored red by the addition of an alkaline solution of
phenolphthalein. When the contents of the tube had gelatinized,
the latter was placed horizontally, connecting two beakers filled
with dilute sulphuric acid as shown in the diagram. A current
of electricity was passed from one electrode A to the other elec-
trode C.

The hydrogen ions from the anode vessel were thus carried along
the tube, and discharged the red color of the phenolphthalein as
they migrated toward the cathode. In this manner the velocity

* Brit. Assoc. Report, p. 393 (1886).

412 THEORETICAL CHEMISTRY

of the hydrogen could be observed under a known potential gra-
dient. The observed and calculated values agree excellently. It
was shown that the velocity of the hydrogen ions suffered almost
no retardation from the high viscosity of the gelatine solution.
Whetham,* in his experiments on ionic velocity, employed two
solutions one of which possessed a colored ion, the progress of the
latter being observed and its velocity determined under unit
potential gradient. For example, consider the boundary line
between two equally dense solutions of the electrolytes AC and
BC, C being a colorless and A a colored ion. When a current
passes through the boundary between the two electrolytes, the
anion C will migrate toward the positive electrode while the two
cations, A and J3, will migrate toward the negative electrode
and the color boundary will move with the current, its speed being
equal to that of the colored ion A . In this way Whetham measured
the absolute velocities of the ions, Cu", C^CV', and Cl'. Ionic
velocities have also been determined by Stcele f who observed
the change in the index of refraction of the solution as the ions
migrated. The accompanying table gives a comparison of the
calculated and observed velocities of some of the ions.

Ion,

Velocity (obs.).

Velocity (calc.).

cm. per sec.
0 0026

cm. per sec.
0 0032

Cu"

0 00029

0 00031

Cl'

0 00058

0 00069

Cr207"

0 00047

0 000473

Conductance and lonization. We have already seen that
solutions of strong acids, strong bases and salts exert abnormally-
great osmotic pressures. According to the molecular theory, this
abnormal osmotic activity has been ascribed to the presence in
the solutions of a greater number of dissolved particles than would
be anticipated from the simple molecular formulas of the solutes.
The ratio of the observed to the theoretical osmotic pressure was
represented, according to van't Hoff, by the factor "i."

* Phil. Trans. A., 184, 337 (1893); 196, 507 (1895).
t Phil. Trans. A., 198, 105 (1902).

ELECTRICAL CONDUCTANCE 413

In 1887, Arrhenius showed that there is an intimate connection
between electrical conductance and abnormal osmotic activity,
only those solutions conducting the electric current which exert
abnormally-high osmotic pressures. It had already been pointed
out by Kohlrausch, that the equivalent conductance of a solution
increases at first with the dilution and then ultimately becomes
constant. Arrhenius explained this behavior by assuming that
the molecules of the solute are dissociated into ions, the con-
ductance of the solution being solely dependent upon the number
of ions present. The dissociation increases with the dilution until
finally, when the equivalent conductance has reached its maximum
value, it is complete, the molecules of solute being entirely broken
down into ions. This theory of Arrhenius, known as the theory
of electrolytic dissociation, is based, as has been pointed out,
upon the views advanced by Clausius. Arrhenius showed how
the degree of dissociation of an electrolyte can be calculated
from the electrical conductance of its solutions. According to the
theory of electrolytic dissociation, the conductance of a solution
is dependent upon the number of ions present in the solution,
upon their charges, and upon their velocities. Since the electric
charges carried by equivalent amounts of the ions of different
electrolytes are equal, and since the velocities of the ions for the
same electrolyte are practically independent of the dilution of the
solution, it follows that the increase in equivalent conductance
with dilution must depend almost wholly upon the increase in the
number of ions present.

The equivalent conductance at infinite dilution has been shown
by the law of Kohlrausch to be

AQO = lc + la,

and, therefore, the equivalent conductance at any dilution v, must
be

At, = a (lc + la),

where a is the degree of dissociation of the electrolyte. Dividing
the second equation by the first, we obtain

Aoo

414

THEORETICAL CHEMISTRY

This equation enables us to calculate the degree of ionization of
an electrolyte at any dilution, provided the conductance of the
solution at the particular dilution is known, together with its
conductance, at infinite dilution. For example, A, at 18° for a
molar solution of sodium chloride is 74.3, and A^ is 110.3; there-
fore, a = 74.3 -5- 110.3 = 0.673, or in a molar solution, the mole-
cules of sodium chloride are dissociated to the extent of 67.3 per
cent. A comparison of the values of i based upon conductance
and osmotic data has already been given in the table on page 230.
Since Av = a (lc + ia), we may also write

The Dissociation of Water. Water behaves as a very weak
binary electrolyte, dissociating according to the equation,

H20 <=> H* + OH'.

The specific conductance of water, purified with the utmost care,
has been determined by Kohlrausch and Heydweiller.* Their
results are given in the following table: —

Temperature,
degrees.

Specific Conduct-
ance xio-«.

0 014

0 040

0 055

0 084

0 170

The conductance of pure water at 0° is so small that one milli-
meter of it has a resistance equal to that of a copper wire of the
same cross-section and 40,000,000 kilometers in length, or in
other words, long enough to encircle the earth one thousand times.
Knowing the specific conductance of water, its degree of dissoci-
ation can be easily calculated. The ionic conductances of the
two ions of water at 18° are as follows: — H* = 318, and OH' =

* Zeit. phys. Chem., 14, 317 (1894).

ELECTRICAL CONDUCTANCE 415

174. Therefore, the maximum equivalent conductance of water

should be

A^ = 318 + 174 = 492.

The equivalent conductance at 18°, of a liter of water between
electrodes 1 cm. apart is, according to the data of Kohlrausch,

0.04 X 10~6 X 103 - 0.04 X 10~3;
therefore

0 04 X 10~3

~ — Zoo ---- = 0-8 x 1Q~7 ^ c> the concentration of the ions,

H* and OH', in mols per liter at 18°.

Conductance of Difficultly-Soluble Salts. In a saturated
solution of a difficultly-soluble salt, the solution is so dilute that in
general we may assume complete ionization, or AB = A^.

When this is the case, we have

^solution

and

Av = A^ = 1000 KV.
Hence

V = AQO •
1000 jc '

or if m denotes the concentration in gram-equivalents per liter,
we have

1 1000 K

Thus, Bottger found for a saturated solution of silver chloride at
20°, K' = 1.374 X 10~6. Deducting the specific conductance of
the water at this temperature, we have

K = 1.374 X 10-6 - 0.044 X 1Q-6 = 1.33 X 1Q-6.

Since the value of A^, at 20°, for silver chloride, determined from
the table of ionic conductances, is 125.5, we have

1000 X 1.33 X 10~6 , AA w ln , . A ni r,

m = - T255 — " — = gr.-equiv. AgCl per liter.

416

THEORETICAL CHEMISTRY

Temperature Coefficient of Conductance. When the temper-
ature of a solution of an electrolyte is raised, the equivalent con-
ductance usually increases. The increase in conductance is due,
not to an increase in the ionization, but to the greater velocity
of the ions caused by the diminution of the viscosity of the solution.
According to Kohlrausch, the relation between conductance and
temperature may be approximately expressed by the following
equation,

A, = Ai8'{l+|S(*- 18)1,

where ft is the temperature coefficient, or change in conductance
for 1° C. Solving the equation for 0, we have

The temperature coefficients of several of the more common elec-
trolytes are given in the accompanying table.

TEMPERATURE COEFFICIENTS OF CONDUCTANCE.

Electrolyte

Nitric acid
Sulphuric acid

Hydrochloric acid

Potassium hydroxide

Potassium nitrate

Potassium iodide

Potassium bromide

Potassium chlorate

Silver nitrate .

Potassium chloride

Ammonium chloride ....
Potassium sulphate
Copper sulphate
Sodium chloride
Sodium sulphate
Zinc sulphate

Temperature
Coefficient.

0 0163
0 0164
0.0165
0 0190
0 0211
0 0212
0 0216
0 0216
0 0216
0 0217
0 0219
0 0223
0.0225
0.0226
0.0234
0 0250

The temperature coefficient of conductance is not, however, a
simple linear function of the temperature. The following empiri-

ELECTRICAL CONDUCTANCE 417

cal equations, expressing equivalent conductance at infinite dilu-
tion at any temperature t in terms of the conductance at 18°, have
been derived by Kohlrausch: —

Aoo* = Aoo18o { 1 + a (t - 18) + ft (t - 18)2},
and

P = 0,0163 (a - 0.0174).

When the values of A»18o, a, and /3, as determined for a large
number of electrolytes, are substituted in the above equation, he
showed that Aoo* becomes equal to zero at a temperature approx-
imating to —40°. Kohlrausch suggested that each ion moving
through the solution carries with it an " atmosphere" of solvent,
and that the resistance offered to the motion of the ion is simply
the frictional resistance between masses of pure water. This
view is in harmony with the solvate theory discussed in an earlier
chapter. Washburn * has calculated the degree of ionic hydration
for several ions. He finds, for example, that the hydrogen ion
carries with it 0.3 molecule of water, while the lithium ion is
hydrated to the extent of 4.7 molecules of water.

Conductance at High Temperatures and Pressures. The
conductance of several typical electrolytes, at temperatures rang-
ing from that of the room up to 306°, have been measured by A. A.
Noyes and his co-workers.f These determinations were made in
a conductance cell especially constructed to withstand high
pressures.

The results show that the values of A«> for binary electrolytes
become more nearly equal with rise of temperature. This may
be taken as an indication of the fact that the ionic velocities tend
to become more nearly equal as the temperature rises. The
conductance of ternary electrolytes increases uniformly with the
temperature, and attains values which are considerably greater
than those reached by binary electrolytes. This is what might
be expected, since if an ion is bivalent, as in a ternary electrolyte,
the driving force is greater, and the ion must move faster, and,
consequently, the conductance must be greater.

* Jour. Am. Chem. Soc., 30, 322 (1909).

t Publication of Carnegie Institution, No. 63.

418 THEORETICAL CHEMISTRY

The temperature coefficient of conductance for binary elec-
trolytes is greater between 100° and 156°, than below or above
these temperatures. The temperature coefficients of ternary
electrolytes increases uniformly with rising temperature. In the
case of acids and bases, the rate of increase in conductance steadily
diminishes as the temperatures rises. The ionization decreases
regularly with rise in temperature, the temperature coefficient of
ionization being small between 18° and 100°. The effect of pres-
sure on conductance was studied by Fanjung.* He found that
the conductance increases slightly with increasing pressure.
This result he interprets as being due to increased ionic velocity
rather than to an increase in the number of ions present in the
solution.

Conductance of Non-aqueous Solutions. A large amount of
interesting and important work has been done in recent years
upon the electrical conductance of solutions in non-aqueous sol-
vents.

It is impossible to give even a brief survey of the results of these
investigations, and we must limit ourselves to the statement of
the following general conclusions : f —

(1) The conditions in non-aqueous solutions are much more
complex than in aqueous solutions.

(2) In general, the laws which have been found to apply to aque-
ous solutions also apply to non-aqueous solutions.

(3) Different solvents appear to have different dissociating
powers.

(4) The dissociating power appears to run parallel with the
dielectric constant of the solvent.

Many interesting phenomena present themselves in connec-
tion with the conductance of electrolytes in mixed solvents, but
for an account of this work the student must consult the original
papers of Jones and his students. J

* Zeit. phys. Chem., 14, 673 (1894).

t "Elektrochemie der nichtwassrigen Losungen," by G. Carrara, Ahren's
'Sammlung Chemischer und chemisch-technischer Vortraege," Vol. XII.
t Publication of the Carnegie Institution, No. 80.

ELECTRICAL CONDUCTANCE

419

Ionizing Power of Solvents. Thomson * and Nernst f pointed
out that if the forces which hold the atoms in the molecule are of
electrical origin, then those liquids which possess large dielectric
constants should have correspondingly great ionizing power.
This is a direct consequence of Coulomb's law of electrostatic
attraction, which may be expressed by the equation,

in which q\ and #2 denote two electric charges, d the distance
between them, / the force of attraction and K the dielectric con-
stant. Obviously the larger K becomes, the smaller will be the
value of/; i.e., the more likely the molecule will be to break down
into ions. That the above relation is approximately true may be
seen from the following table: —

DIELECTRIC CONSTANTS.

Solvent.

Ionizing Power.

Benzene

2 3

Extremely weak

Ethyl ether

4.1

Weak

Ethyl alcohol

Fairly strong

Formic acid .... ....

Strong

Water . .

Very strong

Hydrocyanic acid

Very strong

Dutoit and Aston J have suggested that there is a connection
between the ionizing power of a solvent and its degree of associa-
tion, and Dutoit and Friderich § conclude that the values of AGO,
for a given electrolyte dissolved in different solvents, are a direct
function of the degree of association and an inverse function of
the viscosity of the solvents. Water and the alcohols furnish
good illustrations of the truth of this generalization.

* Phil. Mag., 36, 320 (1893).

t Zeit. phys. Chem., 13, 531 (1894).

j Compt. rend., 125, 240 (1897).

§ Bull. Soc. Chim. [3], 19, 321 (1898).

420

THEORETICAL CHEMISTRY

Conductance of Fused Salts. While solid salts are exceed-
ingly poor conductors of electricity, yet as the temperature is
raised their conductance increases until at their melting-point
they may be grouped with good conductors. There is no sudden
increase in conductance at the melting-point. The specific
conductance of a fused salt may exceed the specific conductance
of the most concentrated aqueous solutions, but owing to the high
concentration the equivalent conductance is much less. The
following table gives the specific and equivalent conductance of
fused silver nitrate: —

Temperature,
degrees

Sp. Corid.

Equiv Cond.

218 (inelt.-pt.)

0 681

29 2

250

0 834

36 1

300

1 049

46 2

350

1.245

55 4

The specific conductance of a 60 per cent aqueous solution of
silver nitrate at 18° is 0.208 reciprocal ohm.

If the salts are impure the conductance is raised, the effect of
impurities being apparent even before the salts have reached their
melting-points. This is analogous to the behavior of solutions,
and suggests that the impurity functions in the salt mixture as a
dissolved solute.*

PROBLEMS.

1. An aqueous solution of copper sulphate is electrolyzed between
copper electrodes until 0.2294 gram of copper is deposited. Before elec-
trolysis the solution at the anode contained 1.1950 grams of copper, after
electrolysis 1.3600 grams. Calculate the transport numbers of the two
ions, Cu" and S04". Ans. n = 0.28, 1 - n = 0.72.

2. A solution containing 0.1605 per cent of NaOH was electrolyzed
between platinum electrodes. After electrolysis 55.25 grams of the
cathode solution contained 0.09473 gram of NaOH, whilst the concen-
tration of the middle portion of the electrolyte was unchanged. In a

* For a complete treatment of fused electrolytes the student is advised to
consult, "Die Elektrolyse geschmolzener Salze," by Richard Lorenz.

ELECTRICAL CONDUCTANCE 421

silver coulometer the equivalent of 0.0290 gram of NaOH was deposited
during electrolysis. Calculate the transport numbers of the Na" and OH'
ions. Ans. n = 0.791, 1 — n = 0.209.

3. In a 0.01 molar solution of potassium nitrate, the transport num-
bers of the cation and anion are, respectively, 0.503 and 0.497. Find the
equivalent conductances of the two ions in this solution having given that
its specific conductance is 0.001044. Ans. lc = 52.5, la = 51.9.

4. The absolute velocity of the Ag* ion is 0.00057 cm. per sec., and that
of the Cl' ion is 0.00069 cm. per sec. Calculate the equivalent con-
ductance of an infinitely dilute solution of silver chloride.

5. The equivalent conductance of an infinitely dilute solution of am-
monium chloride is 130; the ionic conductances of the ions OH' and Cl'
are 174 and 65.44 respectively. Calculate the equivalent conductance
of ammonium hydroxide at infinite dilution. Ans. Aoo = 238.56.

6. The equivalent conductance of a molar solution of sodium nitrate
at 18° is 66; its conductance at infinite dilution is 105.3. What is the
degree of ionization in the molar solution? Ans, a = 62.6 per cent.

7. The specific conductance of a saturated solution of AgCN at 20°
is 1.79 X 10~6 and the specific conductance of water at the same temper-
ature is 0.044 X 10~6 reciprocal ohms. The equivalent conductance at
infinite dilution is 115.5. Calculate the solubility of AgCN in grams per
liter. Ans. 2.02 X 10~3 gram/liter.

8. The equivalent conductance at 18° of a solution of sodium sulphate
containing 0.1 gram-equivalent of salt per liter is 78.4, the conductance
at infinite dilution is 113 reciprocal ohms. What is the value of i for
the solution? What is its osmotic pressure?

Ans. i = 2.388; osmotic pressure = 2.85 atmos.

9. The freezing-point of a 0.1 molar solution of CaCl2 is — 0°.482.
(a) Calculate the degree of ionization (freezing-point constant = 1.89
for one mol per liter), (b) Calculate the degree of ionization from the
equivalent conductance at 18°, which is 82.79 reciprocal ohms, whilst the
equivalent conductance of CaCl2 at infinite dilution is 115.8 reciprocal
ohms. Ans. (a) a = 0.774; (b) « = 0.715.

CHAPTER XIX.
ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS.

Ostwald's Dilution Law. It has been shown in preceding
chapters that the law of mass action is applicable to chemical
equilibria in both gaseous and liquid systems. We now proceed
to show that it applies equally to electrolytic equilibria. When
acetic acid is dissolved in Water it dissociates according to the
equation

CH3COOH <F± CH3COO' + H\

Let one mol of acetic acid be dissolved in water and the solution
diluted to v liters, and let a denote the degree of dissociation.

Then, the concentration of the undissociated acid is - and

the concentration of the ions is — . Applying the law of mass
action, we have

where K is the equilibrium or ionization constant.

This equation expressing the relation between the degree of
ionization and dilution, was derived by Ostwald * and is known

as the Ostwald dilution law. Since a = -—• , we may substitute

Aoo

this value of a in equation (1) and obtain the expression

A,2

-A.)» "

* Zeit. phys. Chem., 2, 36 (1888); 3, 170 (1889).
422

(2)

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 423

The dilution law may be tested by substituting the value of a,
corresponding to any dilution v, in the equation and calculating
the value of the ionization constant, K; the value of a at any
other dilution may then be calculated and compared with the
value determined by direct experiment. The following' table gives
the results obtained with acetic acid at 14°. 1, K being equal to
0.0000178: —

v (in liters)

«X102 (calc).

<*Xl02 (obs.).

0.994

0 42

0 40

2 02

0 60

0 614

15 9

1 67

1 66

18 1

1 78

1.78

1,500 0

15 0

14.7

3,010 0

20.2

20 5

7,480 0

30 5

30 1

15,000 0

40 1

40 8

As will be seen, the agreement between the observed and cal-
culated values is very close. The table also shows to how small
an extent the molecules of acetic acid are broken down into ions,
a molar solution being dissociated less than 0.5 per cent. The
dilution law holds for nearly all organic acids and bases, but fails
to apply to salts, strong acids, and strong bases. When a is
small, the term (1 — a) does not differ appreciably from unity,
and equation (1) becomes

a = VvK.

(3)

On the other hand, when a cannot be neglected, we have, on
Bolving equation (1) for a,

(4)

The method of derivation indicates that the dilution law is
only strictly applicable to binary electrolytes, and therefore,

424 THEORETICAL CHEMISTRY

it is improbable that it will hold for electrolytes yielding more
than two ions. It has been found, however, that organic acids
whether they are mono-, di-, or polybasic always ionize as
a monobasic acid up to the dilution at which a. = 50 per
cent. This means that the dilution law is applicable to poly-
basic acids up to that dilution at which the acid is 50 per cent
ionized.

Strength of Acids and Bases. There are several methods by
which the relative strengths of acids can be estimated. A method
which has proved of great value is that in which two different
acids are allowed to compete for a certain base, the amount of
which is insufficient to saturate both of them. Suppose equiva-
lent weights of nitric and dichloracetic acids together with sufficient
potassium hydroxide to saturate one acid completely are taken:
we then determine the position of the equilibrium represented by
the equation

HN03 + CHC12 • COOK *± CHC12 • COOH + KN03.

In order to determine the conditions of equilibrium we may make
use of any method which does not disturb this equilibrium. Since
ordinary chemical methods are excluded on this account, we
employ any physical property which is capable of exact measure-
ment and differs sufficiently in the two systems, as for example,
the change in volume, or the thermal change, accompanying
neutralization. Thus, Ostwald * found that when one mol of
potassium hydroxide is neutralized by nitric acid in dilute solu-
tion, the volume increases approximately 20 cc. When one mol
of potassium hydroxide is neutralized by dichloracetic acid, how-
ever, the increase in volume is 13 cc. It is evident, therefore, that
if nitric acid completely displaces dichloracetic acid as represented
by the above equation, the increase in volume will be 20 — 13
= 7 cc.; if no displacement occurs, then the volume will remain
constant. He found that the volume actually increased 5.67 cc.
Therefore, the reaction represented by the upper arrow has pro-
ceeded to the extent of 5.67 •*- 7 = 80 per cent. That is to say,

* Jour, prakt. Chem. [2], 18, 328 (1878).

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 425

in the competition of the two acids for the base, the nitric acid
has taken 80 per cent and the dichloracetic acid has taken 20 per
cent, or the relative strengths of the two acids are in the ratio of
80 : 20, or 4 : 1.

The relative strengths of acids can also be determined from their
catalytic effect on the rates of certain reactions, such as the
hydrolysis of esters or the inversion of cane sugar.

The order of the activity of acids is the same whether measured
by equilibrium or kinetic methods. Arrhenius pointed out that
the relative strengths of acids can be readily determined from
their electrical conductance. The order of the strengths of acids
as determined by equilibrium and kinetic methods is the same as
that of their electrical conductances in equivalent solutions.
This is well illustrated by the following table in which the three
methods are compared, hydrochloric acid being taken as the
standard of comparison: —

Acid.

Method Employed.

Equilibrium.

Kinetic.

Conductance.

HC1

100
100
49
9

100
100
53.6
4.8
0.4

100
99.6
65.1
4.8
1.4

HNOs

H2SO4

CH2C1COOH

CHsCOOH

The results of these and other experiments warrant the con-
clusion that the strength of an acid is determined by the number
of hydrogen ions which it yields. It is important to note that the
electrical conductance of an acid is not directly proportional to
its hydrogen ion concentration; the relatively high velocity of
the H ion is the cause of the approximate proportionality between
these two variables. In the case of a weak acid, the value of the
ionization constant may be taken as a measure of the strength of
the acid. The following table gives the values of the ionization
constants at 25° for several different acids.

426

THEORETICAL CHEMISTRY
IONIZATION CONSTANTS OF ACIDS.

Acid.

lonization
Constant.

Acetic acid

0 0000180

Monochloracetic acid ....
Trichloracetic acid
Cyanacetic acid
Formic acid

0.00155
1.21
0 0037
0 000214

Carbonic acid
Hydrocyanic acid .

3040X10-10
570X10~10

Hydrogen sulphide ...
Phenol .

13X10-10
1 3X10~10

Since for a weak acid, a = vWf, it follows that for two weak
acids at the same dilution, we may write

or the ratio of the degrees of ionization of the two acids is equal to
the square root of the ratio of their ionization constants. Thus,
from the data given in the foregoing table for acetic and mono-
chloracetic acids, we have

ai=1/0.
«2 V 0.

0.000018

_
00155 9.3'

or the effect of replacing one atom of hydrogen in the methyl
group of acetic acid increases the strength of the acid about nine
times.

Just as the hydrogen ion concentration of acids determines their
strength, so the strength of bases is determined by the concen-
tration of hydroxyl ions. The strength of bases may be estimated
by methods similar to those employed in determining the strength
of acids. Thus, two different bases may be allowed to compete
for an amount of acid sufficient to saturate only one of them; or
a catalytic method developed by Koelichen * may be used. This
method is based upon the effect of hydroxyl ions on the rate of
condensation of acetone to diacetonyl alcohol, as represented by
the equation

2CH3COCH3 - CH3COCH2C (CH3)2OH.
* Zeit. phys. Chem., 33, 129 (1900).

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 427

In addition to these two methods, the method of electrical con-
ductance is also applicable. The agreement between the results
obtained by the three methods is quite satisfactory. The alkali
and alkaline earth hydroxides are very strong bases and are dis-
sociated to about the same extent as equivalent solutions of
hydrochloric and nitric acids, while on the other hand, ammonia
and many of the organic bases are very weak. The following
table gives the ionization constants of several typical bases: —

IONIZATION CONSTANTS OF BASES.

Base.

Ionization
Constant.

Ammonia

0 000023

Methyl amine .

0 00050

Trimcthylamine

0 000074

Pyridine

2 5X10~JO

Aniline

1.1X10"10

Mixtures of Two Electrolytes with a Common Ion. Just as
the dissociation of a gaseous substance is diminished by the addi-
tion of an excess of one of the products of dissociation, so the
ionization of weak acids and bases is depressed by the addition of
a salt with an ion common to the acid or the base. If the degree
of ionization of a salt with an ion in common with an acid or a
base is represented by a', and n denotes the number of molecules
of salt present, then the equation of equilibrium of the acid or
base will be

(not + a} a = Kv (1 - a),

where a is the degree of ionization of the acid or base. For very
weak acids and bases, a is so small that 1 — a does not differ
appreciably from unity, and since a! is practically independent of
the dilution, we obtain

na = Kv
or

Kv
a = — •

428 THEORETICAL CHEMISTRY

That is, the ionization of a weak acid or base, in the presence of
one of its salts, is approximately inversely proportional to the
amount of salt present.

In many of the processes of analytical chemistry, advantage is
taken of the action of neutral salts on the ionization of weak acids
and bases. Thus, while the concentration of hydroxyl ions in
ammonium hydroxide is sufficient to precipitate magnesium hy-
droxide from solutions of magnesium salts, the presence of a small
amount of ammonium chloride depresses the ionization of the
ammonium hydroxide to such an extent that precipitation no
longer takes place.

Isohydric Solutions. Arrhenius * was the first to point out
what relation must exist between solutions of two electrolytes
with a common ion, in order that, when mixed in any proportions,
they may not exert any mutual influence. He showed that when
the concentration of the common ion in each of the two solutions
is the same before mixing, no alteration in the degree of ionization
will occur after mixing. Such solutions are said to be isohydric.
Thus, an aqueous solution containing one mol of acetic acid in
8 liters, is isohydric with an aqueous solution containing one mol
of hydrochloric acid in 667 liters. On mixing these two solutions
the hydrogen ion concentration remains unchanged, and if the
mixture is treated with a small amount of sodium hydroxide,
equal amounts of sodium acetate and sodium chloride will be
formed.

That isohydric solutions may be mixed without altering their
respective ionizations may be shown in the following manner: —
Let C and c denote the concentrations of the undissociated por-
tions, and CA, C2, CA, and C2 denote the concentrations of the dis-
sociated portions of two electrolytes, and let C2 and 02 correspond
to two different ions.

Then, we have

kc = CA<%, (1)

and

KC = CUC,. (2)

* Wied. Ann., 30, 51 (1887).

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 429

If v liters of the first solution be mixed with V liters of the second
solution, the concentrations of the undissociated portions, and of
the dissimilar ions, will be

Cv cv C^V , c^v

jf+V v~+^> T+v' T+v*

while the concentration of the common ion A, becomes

Applying the law of mass action, we have

JL--
fee-

and

But equations (3) and (4) only become identical with equations (1)
and (2) when C^ = cAj or, in other words, no change in the degree
of dissociation takes place after the two solutions are mixed.

lonization of Strong Electrolytes. It has already been men-
tioned that the Ostwafd dilution law, which is a direct conse-
quence of the law of mass action, applies only to weak electrolytes.
Just why the law of mass action should fail to apply to strong
electrolytes is not known, but several possible causes have been
suggested to account for its failure. One of the most plausible
explanations is that advanced by Biltz,* who attributes the
failure of the law of mass action when applied to strong electrolytes,
to hydration of the solute. If the ions become associated with
a large proportion of the solvent, the effective ionic concentration
would then be the ratio of the amount of the ion present to that
of the free solvent, instead of to the total solvent, as ordinarily
calculated. This view is in harmony with certain facts which
have been adduced in favor of the theory of solvation. While
the Ostwald dilution law does not apply to strongly ionized elec-
trolytes, certain empirical expressions have been derived which

* Zeit. phys. Chem., 40, 218 (1902).

430

THEORETICAL CHEMISTRY

hold fairly well over a wide range of dilution,
showed that the equation

_ rrt

(1 - a) Vt) '

Thus, Rudolphi

gives approximately constant values for K ' for strong electrolytes
The following table gives the results obtained with solutions o
silver nitrate at 25°; the numbers in the third column being cal
culated by means of the Ostwald dilution law, while those in th
fourth column are calculated by means of Rudolphi's dilution law

0 8283

0 253

1.11

0 8748

0 191

1 16

0 8993

0 127

1.06

128

0 9262

0 122

1 07

256

0 9467

0 124

1 08

512

0 9619

0 125

1 09

The Rudolphi equation was modified by van't Hofff to th<
form

This equation holds even more closely than that of Rudolphi
Of the more recent empirical equations which have been derivec
to express the change of conductance of an electrolyte with dilu-
tion, the equations of Kraus * and Bates t deserve special men-
tion.
The equation of Kraus has the following form: —

/ Aw \2 C 7 , , ,

\AoW A _ A?;^

AOW

In this equation A is the conductance of the electrolyte whose con-
centration C is expressed in mols per liter, TJ/TJO is the ratio of the
viscosity of the solution to that of the solvent, and k, kf, h, and AQ

* Jour. Am. Chem. Soc., 35, 1412 (1913).
t Ibid., 37, 1421 (1915).

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 431

are empirical constants the values of which are so chosen as to
insure close agreement between the observed and calculated values
of the conductance.

The equation of Bates is similar to that of the preceding equa-
tion, except that the logarithm of the left-hand side of the equation
is substituted for the original expression of Kraus. The equation
of Bates takes the form : —

l°&o TI-

i _ f A?? ^

\AoW

The constants fc, fc', and h are purely empirical as in the equation
of Kraus, but A0 denotes the equivalent conductance at infinite
dilution.

A comparison between the two equations is afforded by the fol-
lowing table in which is recorded the observed and calculated
values of the " corrected " equivalent conductance, Aiy/r/o, for
solutions of potassium chloride at 18°.

COMPARISON OF THE EQUATIONS OF KRAUS AND BATES.

•n/'-no

A (obs.)

Af/170

AWlo (K.)

A*/™ (B.)

0 9954

88.3

87.89

87.4

89.3

0 9805

92 53

90 73

90 9

91 9

0 982

98 22

96 5

96 4

96 53

0 5

0 9898

102 36

101 32

101.1

101 29

0 2

0 9959

107 90

107 46

107.6

107 43

0 1

0 9982

111 97

111 77

111 9

111 73

0 05

0 9991

115 69

115 59

115 5

115 58

0 02

0 9996

119 90

119 85

119.8

119 83

0 01

0 9998

122 37

122 35

122 4

122 32

0 005

0 9999

124 34

124 33

124 4

124 38

0.002

1.0000

126 24

126 3

126 31

0 001

127 27

127.27

127 2

127 32

0 0005

128 04

127.6

128 05

0 0002

128 70

127 9

128.68

0 0001

129 00

128.1

128.96

0 0

129 50

128 3

129.50

It will be observed that for dilute solutions, the ratio 17/170 is
practically unity and, furthermore, that the value of CA/Ao is so
small that the second term on the right-hand side of both equations
is negligible in comparison with the value of k.

432

THEORETICAL CHEMISTRY

Therefore, under these conditions, both equations reduce to the
form

2 . .

= constant,

which will be recognized as identical with Ostwald's dilution law
as given on page 422.

Heat of Ionization. The heat of ionization of an electrolyte
can be calculated by means of the reaction isochore equation of
van't Hoff (see p. 324), provided the degree of ionization at two
different temperatures is known.

Since I

and

(l-ajv'

it follows that the heat of ionization may be calculated by means
of the equation

2.3026 R \ log

' (1 -

-log

(1 - ««) V

Arrhenius * has shown that this equation also applies to those
electrolytes which do not obey the Ostwald dilution law. Some
of the results obtained by Arrhenius are given in the accompany-
ing table: —

Electrolyte

Temperature.

Calories.

Acetic acid

35°

386

Propi'onic acid

21°. 5
35°

-28
557

Butyric acid

i
J

21°. 5
35°

183
935

Phosphoric acid

I
\

21°. 5
35°

427
2458

Hydrochloric acid

21°. 5
35°

2103
1080

Potassium chloride

35°

362

Potassium bromide

35°

425

Potassium iodide

35°

916

Sodium chloride . . . ."

35°

454

Sodium hydroxide

35°

1292

Sodium acetate

35°

391

* Zeit. phys. Chem., 4, 96 (1889).

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 433

It will be found that the values of the heats of ionization given
in this table do not agree with the values calculated for these
same substances from the data given in the table on page 307.
The reason for this lack of agreement is, that the data of the earlier
table refer to the heat of formation of the ions from the dissolved
substance, whereas the data of the table just given represent the
combined thermal effects of solution and ionization.

The Solubility Product. While the law of mass action does
not in general apply to the equilibrium between the dissociated
and undissociated portions of an electrolyte, — except in the case
of organic acids and bases, — it does apply with a fair degree of
accuracy to saturated solutions of electrolytes.

A saturated solution of silver chloride affords an example of
such an equilibrium. This salt is practically completely ionized
in a saturated solution, as represented by the equation

Applying the law of mass action to this equilibrium, we obtain

CAg- X CGI TT

- = A.

CAgCl

Since the solution is saturated, the value of CAKCI must remain
constant at constant temperature, and therefore

£Ag* X ccr = constant = s,

where the product of the ionic concentrations s is called the solu-
bility or ionic product.

The equilibria in the above heterogeneous system may be repre-
sented thus: —

(in solution) (solid)

The solubility product for silver chloride at 25° is 1.56 X 10~10,
the ionic concentrations being expressed in mols per liter. Hence,
since the two ions are present in equivalent amounts, a saturated
solution of silver chloride at 25° must contain Vl.56 X 10~10
= 1.25 X 10~5 mols per liter of Ag* and Cl' ions. In general, if

434 THEORETICAL CHEMISTRY

represents the equilibrium between an electrolyte and its products
of dissociation in saturated solution, we have

The solubility product may be defined as the maximum product of
the ionic concentrations of an electrolyte which can exist at any one
temperature.

Just as the dissociation of a gaseous substance or of an organic
acid is depressed by the addition of one of the products of dis-
sociation, so when a substance with a common ion is added to the
saturated solution of an electrolyte, the dissociation is depressed
and the undissociated substance is precipitated.

The following example will serve to illustrate how the solu-
bility product of a substance can be determined, and how the
change in solubility due to the addition of a substance containing
a common ion may be calculated. The solubility of silver bromate
at 25° is 0.0081 mol per liter. If we assume complete ionization,
the concentration of the ions, Ag* and Br(V will be the same
and equal to 0.0081 mol per liter, or

(0.0081) (0.0081) = s.

The solubility in a solution of silver nitrate containing 0.1 mol
of Ag' ions can be calculated from the equation

(0.0081)2 = (0.0081 + 0.1 - x) (0.0081 - x),

where x represents the amount of silver bromate thrown out of so-
lution by the addition of 0.1 mol of Ag* ion. Since (0.0081 — x)
represents the concentration of BrO3' ions after the addition of
the silver nitrate, it also represents the solubility of silver bromate
under similar conditions. The effect of adding a solution of a
soluble bromate containing 0.1 rnol of Br03' ion will be the same
as that produced by 0.1 mol of Ag' ion.

The Basicity of Organic Acids. The Ostwald dilution law
holds strictly for all monobasic organic acids, and also for poly-
basic organic acids which are less than 50 per cent ionized. The
neutral salts of these acids, however, are much more highly ionized,
and the difference in conductance between two dilutions of a neu-
tral salt of a polybasic acid is greater than the difference in

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 435

conductance between the same dilutions of a neutral salt of
a monobasic acid. Ostwald * has shown that it is possible
to estimate the basicity of an organic acid from the difference
in the equivalent conductance of its sodium salt at two different
dilutions.

As the result of a long series of experiments, he found that the
difference between the equivalent conductance of the sodium salt
of a monobasic organic acid at v = 32 liters and at v = 1024 liters
is approximately 10 units. Similarly, the difference for a dibasic
acid between the same dilutions is 20 units, and for an w-basic
acid the difference is 10 n. Hence, to estimate the basicity of an
organic acid, the equivalent conductance of its sodium salt at v =
32 liters and at v = 1024 liters is determined; then, if A is the dif-
ference between the values of the conductance at the two dilutions,

the basicity will be n = -^ -

The following table gives the values of A and n for the sodium
salts of several typical organic acids: —

Acid.

Formic

10 3

Acetic

9 5

Propionic

10 2

Benzole

8 3

Quininic

19 8

Pyridine-tricar boxy lie (1, 2, 3)

31 0

Pyridine-tricarboxylic (1, 2, 4) . ...

29.4

Pyridine-tetracarboxylic . .

41 8

Pyridine-pentacarboxylic

50 1

Influence of Substitution on lonization. Attention has already
been called to the marked difference in the strength of acetic
acid produced by the replacement of the hydrogen atoms of the
methyl group by chlorine. In the accompanying table the ioniza-
tion constants for various substitution products of acetic acid
are given: —

* Zeit. phys. Chem., i, 105 (1887); 2, 902 (1888).

436

THEORETICAL CHEMISTRY

Acid.

lonization
Constant (25°).

Acetic. CHsCOOH

0 000018

Propionic, CH8CH2COOH

0 000013

Chloracetic, CH2C1COOH

0 00155

Bromacetic, CH2BrCOOH.. .

0 00138

Cyanacetic, CH2CNCOOH

0 00370

Glycollic, CH2OHCOOH

0.000152

Phenylacetic, C6H6CH2COOH

0.000056

Amidoacetic, CH2NH2COOH

3 4 X 10~10

This table affords an interesting illustration of the influence of
different substituents on the strength of acetic acid. Thus, the
activity of the acid is increased by the replacement of alkyl hydro-
gen atoms by Cl, Br, CN, OH, or C6H5, while the substitution of
the CH3 or NH2 groups diminishes its activity. If we assume
that the substituents retain their ion-forming capacity on enter-
ing into the molecule of acetic acid, these differences in activity
can be readily explained. Thus, Cl, Br, CN, and OH tend to
form negative ions, and hence increase the negative character of
the group into which they enter. On the other hand, basic groups,
such as NH2, diminish the tendency of the group into which they
enter to yield negative ions.

The influence of an alkyl residue on the strength of an organic
acid is conditioned by its distance from the carboxyl group. This
is well illustrated by the ionization constants of propionic acid
and some of its derivatives.

Acid.

Ionization
Constant (26°).

Propionic acid, CH8CH2COOH

0.0000134 .

Lactic acid, CHaCHOHCOOH

0.000138

0-oxypropionic acid, CH2OHCH2COOH

0.0000311

The effect of the OH group in the a-position is seen to be much
more marked than when it occupies the 0-position.

The position of a substituent in the benzene nucleus exerts a
marked influence on the strength of the derivatives of benzoic

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 437

acid. The ionization constants of benzoic acid and the three
chlorbenzoic acids are given in the following table: —

Acid.

Ionization
Constant (25°).

Benzoic acid, C6H6COOH

0 000073

o-Chlorbenzoic acid, C6H4C1COOH

0.00132

m-Chlorbenzoic acid, CeH^ClCOOH

0 000155

p-Chlorbenzoic acid CeH^ClCOOH

0 000093

When the halogen enters the ortho-position, the strength of the
acid is greatly augmented, while in the meta- and para-positions
the effect is much smaller, meta-chlorbenzoic acid being stronger
than para-chlorbenzoic acid. It is a general rule that the influence
of substituents is always greatest in the ortho-position, and least
in the meta- and para- positions, the order in the two latter being
uncertain.

Hydrolysis. When a salt formed by a weak acid and a strong
base, such as sodium carbonate, is dissolved in water, the solution
shows an alkaline reaction, while on the other hand, when a salt
formed by a strong acid and a weak base, such as ferric chloride,
is dissolved in water, the solution shows an acid reaction.

The process which takes place in the aqueous solution of a salt,
causing it to react alkaline or acid, is termed hydrolysis or hydro-
lytic dissociation. If MA represents a salt, in which M is the basic
and A is the acidic portion, then the hydrolytic equilibrium may
be represented by the equation

MA + H20 <± MOH + HA.

If the base formed is insoluble or undissociated and the acid is
dissociated, the solution will react acid. If the acid formed is
insoluble or undissociated and the base is dissociated, the solution
will react alkaline. Finally, if both base and acid are insoluble
or undissociated, the salt will be completely transformed into base
and acid, and, as there will be no excess of either H* or OH' ions,
the solution will remain neutral.

It is evident, then, that hydrolysis is due to the removal of
either one or both of the ions of water by the ions of the salt to

438 THEORETICAL CHEMISTRY

form undissociated or insoluble substances. As fast as the ions
of water are removed, the loss is made good by the dissociation of
more water, until eventually a condition of equilibrium is estab-
lished. The conditions governing hydrolytic equilibrium may be
determined from a knowledge of the solubility or ionic constant
of the substances involved. Thus, if the product of the concen-
trations of the ions M" and OH' exceeds that which can exist in
pure water, then some undissociated or insoluble substance will
be formed. This will disturb the equilibrium of H" and OH'
ions, and a further dissociation of water must occur until the
ionic product of water is just reached.

If now the ions H* and A7 do not unite to form undissociated
acid, the presence of an excess of H* ions will disturb the equi-
librium between pure water and its products of dissociation; or,
since

CH- X COH' = «H2o,

the concentration of OH' ions present, when CH- represents the

total concentration of H" ions, will be - •

CH«

A similar readjustment will take place when an undissociated
or insoluble acid and a dissociated base are formed.

We may now proceed to consider three different cases of hydroly-
sis, viz., when the reaction is caused (1) by the base, (2) by the
acid, and (3) by both base and acid.

CASE I. The formation of an undissociated or insoluble base is pri-
marily the cause of the hydrolysis, the acid formed being dissociated.

Let the hydrolytic equilibrium be represented by the equation

MA + H20<:±MOH + HA.

The reaction will proceed in the direction of the upper arrow until
the product, CM* X COH', exceeds that which can exist in the ab-
sence of an undissociated base. When equilibrium is established,
we have

final CM- X final COH' = KUOH X CMOH formed, (1)

or if the base formed is practically insoluble, the equilibrium equa-
tion simplifies to the form

final CM- X final COH' = «MOH, (2)

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 439

where SMOH is the solubility product of the base. The condition
of equilibrium represented by the equation

CH* X COH' = $H2o

must* be fulfilled. It follows that the final concentration of the
OH' ions will be the quotient obtained by dividing the ionic product
for water, at the temperature of the experiment, by the final con-
centration of the H* ion, this latter being wholly dependent upon
the extent of the reaction and the degree of ionization of the acid
formed. If the degree of hydrolysis of the salt be represented by
x, and the degree of dissociation of the unhydrolyzed portion of
the salt be denoted by «„ then, if one mol of salt be dissolved
in V liters of solution, the final concentration of M" ions will be

— — y - and the final concentration of the undissociated base

X />•

will be y . The total acid formed will be y , and if aa denotes

the degree of dissociation of the acid, the concentration of the H*

y
ions will be <xay~ Substituting these values in equations (1) and

(2), we obtain

, (3)

and

Simplifying equations (3) and (4), we have

(T^xyv ' ^7= K^ = K* (5)

and

(1 - x) a,

From equations (5) and (6) it appears that the constant of hydrol-
ysis can be found either from the ionic product for water and the

440 THEORETICAL CHEMISTRY

ionization constant of the base, or from the ionic product for
water and the solubility product of the base. Furthermore, if the
base formed is insoluble, equation (6) shows that the degree of
hydrolysis, z, is independent of the dilution of the salt, V.

CASE II. The formation of an undissociated or insoluble add
is primarily the cause of the hydrolysis, the base formed being dis-
sociated. In this case hydrolysis takes place until the product
CH' X CA' exceeds that which can exist in the absence of undis-
sociated acid. When equilibrium is established, we have

final CH« X final CA' = ^HAX CHA formed, (7)

or if the acid formed is practically insoluble, the equilibrium equa-
tion simplifies to the form

final CH- X final CA' = SHA» (8)

Since the final CH» = SHZO -*- final COH', we have, final CA' = y ,

oc
final COH' = <*b -y , where ah is the degree of dissociation of the base

formed, and the final CHA = y • Substituting these values in equa-
tions (7) and (8), we obtain

and

-"-*) «-*»_... (10)

Simplifying equations (9) and (10), we have

-x a,
and

(1 — x) a, SHA

It is evident from equations (11) and (12), that the constant of
hydrolysis can be found either from the ionic product for water and

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 441

the ionization constant of the acid, or from the ionic product for
water and the solubility product of the acid.

CASE III. The formation of an acid and a base, both being
slightly dissociated, is the cause of the hydrolysis.

In this case let us assume that K HA is smaller than KMOH.

Since the final cOir = K™°* XCMQH, and since both HA and MOH

CM*

are slightly dissociated, we may write CHA = CMOH = |F, and

a9 (1 - x)
— ^- — -

Substituting these values in equation (7), we obtain

<** 0 ~ x) sH2o „ x , Q,

- y --- — = XHA X y- (13)

V j-r X V

A MOH y

V
Simplifying equation (13), we obtain

_ _

(1 - xY ~a* KJUL X KMOH

From equation (14) we see that the constant of hydrolysis can be
found from the ionic product for water and the ionization constants
of the acid and the base. If both acid and base are practically
insoluble, the reaction will be complete at all dilutions.

As an illustration of the application of the foregoing equations,
we may take the calculation of the degree of hydrolytic dissociation
of potassium cyanide in 0.1 molar solution at 25°. Potassium
cyanide being a salt of a weak acid, the degree of hydrolysis can
be calculated by means of the equation

(l-x)V'a.~K^~ k"

Since at 25° C, KHA = 7.2 X 1Q-10 and sn,o = 1.05 X 10~7)2, we
have

(1.05 X 10-')2

K -
*

7.2 X 10

-10 '

442 THEORETICAL CHEMISTRY

and since in dilute solution aa = o& = 1, we have

s2 (LOS X 10~y)2

(1 - x) 10 7.2 X 10-10 '
or

x = 0.0123

Experimental Determination of Hydrolysis. The degree of
hydrolysis can be determined experimentally in several different
ways. A very convenient method is that based upon measure-
ments of electrical conductance. When a salt reacts hydrolyti-
cally with one mol of water, the limiting value of its equivalent
conductance will be A^ + A#, where A^ and A# denote the equiv-
alent conductances of the acid and base formed. If A is the equiv-
alent conductance of the unhydrolyzed salt, and Ah is the actual
conductance of the salt at the same dilution, then the increase in
conductance corresponding to a degree of hydrolysis x will be
Afc — A, The value of A may be found by determining the
conductance of the salt in the presence of an excess of one of the
products of hydrolysis and deducting from it the conductance of
the substance added. Since if the hydrolysis were complete, the
equivalent conductance would be A^ + A# — A, we have

AA+AB- A

all conductances being measured at the same dilution and the
same temperature. The following example will illustrate the
use of this equation: — At 25°, the equivalent conductance of an
aqueous solution of aniline hydrochloride is 118.6, the dilution
being 99.2 liters. The equivalent conductance in the presence of
an excess of aniline is 103.6, while the equivalent conductance of
hydrochloric acid at the same dilution is 411. The conductance
of pure aniline is so small as to be negligible. Substituting these
values in the equation, we find

118.6-103.6 ArvlQQ
* = 411 - 103.6 - °-0488-

Lunden * has shown how this method may be extended to cases
where both acid and base are slightly dissociated.
* Jour. chim. phys., 5, 145, 574 (1907).

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 443

The lonization Constant of Water. One of the most accurate
methods known for the determination of the ionization constant
of water is based upon measurements of the degree of hydrolytic
dissociation of different salts. Thus Shields* found that a 0.1
molar solution of sodium acetate is 0.008 per cent hydrolyzed at
25°. We may consider the salt, as well as the sodium hydroxide
formed from its hydrolysis, to be completely dissociated at this
dilution. The ionization constant of the acetic acid formed is
0.000018 at 25°. Solving equation (11) (on page 440) for sn2o,
and remembering that «, = «& = 1, we have

Substituting the above values in this expression, we obtain

SH,o - 0.000018 . . 10 - L16 X

and since the ions, H* and OH', are present in equivalent amounts
we have _

CH* = COIF = Vl.16 X 10"14 = 1.1 X 10-7mol per liter.

Kohlrausch obtained from his measurements of the conductance
of pure water at 25°, CH» = COH' = 1.05 X 10~7 mol per liter (see
p. 415).

PROBLEMS.

1. At 25° the specific conductance of butyric acid at a dilution of
64 liters is 1.812 X 10~4 reciprocal ohms. The equivalent conductance
at infinite dilution is 380 reciprocal ohms. What is the degree of ioniza-
tion and the concentration of H" ions in the solution? What is the ioni-
zation constant of the acid?

Ans. a = 0.0305, CH- = 4.765 X 10~4 mol per liter, K = 1.5 X 10-*.

2. The heat of neutralization of nitric acid by sodium hydroxide is
13,680 calories, and of dichloracetic acid, 14,830 calories. When one
equivalent of sodium hydroxide is added to a dilute solution containing
one equivalent of nitric acid and one equivalent of dichloracetic acid,
13,960 calories are liberated. What is the ratio of the strengths of the
two acids? Ans. HN08 : CHC12COOH :: 3.1 : 1.

* Zeit. phys. Chem., 12, 167 (1893).

444 THEORETICAL CHEMISTRY

3. For potassium acetate we have the following data: —

At)(l8°)

67 1

78 4

100

87 9

10000

91 9

and Zoo = 64.67, and 7oo = 35. Compare the constants obtained by the

K' CH3COO'

Ostwald, Rudolphi, and van't Hoff dilution laws.

4. The ioriization constant of a 0.05 molar solution of acetic acid is
0.0000175 at 18°, and 0.00001624 at 52°. Calculate the heat of ionization
of the acid. To what temperature does this value correspond?

Ans. 416 calorics at 35°.

5. At 20° the specific conductance of a saturated solution of silver
bromide was 1.576 X 10~6 reciprocal ohms, and that of the water used
was 1.519 X 10~6 reciprocal ohms. Assuming that silver bromide is
completely ionized, calculate the solubility and the solubility product of
silver bromide, having given that the equivalent conductances of potas-
sium bromide, potassium nitrate, and silver nitrate at infinite dilution
are 137.4, 131.3, and 121 reciprocal ohms respectively.

Ans. CAgBr = 4.49 X 10~7 mol per liter, SAgBr = 2.02 X 10~13.

6. The solubility of silver cyanate at 100° is 0.008 mol per liter. Cal-
culate the solubility in solution of potassium cyanate containing 0.1 mol
of K* ions. Ans. 6.4 X 10"4 mol per liter.

7. Calculate the degree of hydrolytic dissociation of a 0.1 molar solu-
tion of ammonium chloiide, having given the following data: — aa = 0.86,
Q« = 0.87,#NH4OH = 0.000023, and «H2o = (0.91 X 10-7)2at25°.

Ans. x = 0.006 per cent.

8. In the reaction represented by the equation

MA3 + 3 H20 = M (OH), + 3 HA,

the base formed is insoluble. Derive an expression for the constant of
hydrolysis.

Ans. Kp • n

SM(OH)8 (1 —

9. The equivalent conductance of aniline hydrochloride at a dilution
of 197.6 liters is 126.7 reciprocal ohms, at 25°. The equivalent con-

ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 445

ductance of aniline hydrochloride in the presence of an excess of aniline
is 106.6; and the equivalent conductance of hydrochloric acid at the
same dilution is 415. If the conductance of pure aniline is negligible,
calculate the degree of hydrolytic dissociation and the constant of hydrol-
ysis, assuming «, = «0 = 1.

Ans. x = 6.52 per cent, Kh = 2.30 X 10~5.

10. The hydrolysis constant of aniline is 2.25 X 10~6, and the ioniza-
tion constant is 5.3 X 10~10. Calculate the concentration of the H*
and OH' ions in water. Ans. CCH* = CCH' = 1.09 X 10~7.

CHAPTER XX.
ELECTROMOTIVE FORCE.

Galvanic Cells. Since the year 1800, when Volta invented
his electric pile, many different forms of galvanic cell have been
introduced.

It is not our purpose to give a detailed account of these cells,
but rather to give a brief outline of the theories which have been
advanced in explanation of the electromotive force developed in
such cells. When two metallic electrodes are immersed in a solu-
tion of an electrolyte, a current will flow through a wire connect-
ing the electrodes, provided the two metals are dissimilar, or that
a difference exists between the solutions surrounding the electrodes.
An electric current can be obtained from a combination of two
different metals in the same electrolyte, from two different metals
in two different electrolytes, from the same metal in different elec-
trolytes, or from the same metal in two different concentrations of
the same electrolyte.

In order that the electromotive force of the combination shall
remain constant, it is necessary that the chemical changes involved
in the production of the current shall neither destroy the difference
between the electrodes, nor deposit upon either of them a non-
conducting substance. A galvanic combination which fulfils
these conditions very satisfactorily is the Daniell cell. This cell
consists of zinc and copper electrodes immersed in solutions of
their salts, as represented by the scheme

Zn - Sol. of ZnS04 1| Sol. of CuS04 - Cu,

in which the two vertical lines indicate a porous partition separat-
ing the two solutions. When the zinc and copper electrodes are
connected by a wire, a current of positive electricity passes from
the copper to the zinc along the wire. Zinc dissolves from'the zinc
electrode, an equivalent amount of copper being displaced from

446

ELECTROMOTIVE FORCE 447

the solution and deposited simultaneously on the copper electrode.
As long as only a moderate current flows through the cell, the
original nature of the electrodes is not modified, the only change
which occurs being the gradual dilution of the copper sulphate,
owing to the separation of copper and its replacement by zinc.
If the loss of copper sulphate is replaced, the electromotive force
of the cell will remain constant. If, after the cell is assembled no
current be allowed to flow, the copper sulphate will slowly diffuse
into the solution of zinc sulphate, and metallic copper will ulti-
mately be deposited on the zinc electrode. In this way miniature,
local galvanic cells will be formed on the surface of the z

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